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Here is the tested code, you can use it as a mu_plugin:

<?php
/**
 * Filters whether to short-circuit default header status handling.
 *
 * Returning a non-false value from the filter will short-circuit the handling
 * and return early.
 *
 * @param bool     $preempt  Whether to short-circuit default header status handling. Default false.
 * @param WP_Query $wp_query WordPress Query object.
 *
 * @return bool
 * @since 4.5.0
 */
function pre_handle_404_filter( bool $preempt, WP_Query $wp_query ) {
    global $parsed_address, $api_results;

    $parsed_address = get_parsed_address();

    if ( ! $parsed_address ) {
        return $preempt;
    }

    // Make request to  API, get status.
    $status      = true; // For debug purposes.
    $api_results = null;

    if ( $status ) {
        // Save results to $api_results.
        // Return true to avoid 404 processing by WP.
        return true;
    }

    return $preempt;
}

add_filter( 'pre_handle_404', 'pre_handle_404_filter', 10, 2 );

/**
 * Handle 404.
 */
function handle_404() {
    global $parsed_address, $api_results, $wp_query;

    if ( ! $parsed_address ) {
        return;
    }

    // Set 404 here to prevent bugs in get_header().
    $wp_query->set_404();

    get_header();
    echo 'API Results'; // Output $api_results here.
    get_footer();
    exit();
}

add_action( 'template_redirect', 'handle_404', - PHP_INT_MAX );

/**
 * Get parsed address from the request uri.
 * Make sure we have an address in the uri, otherwise return empty string.
 *
 * @return string
 */
function get_parsed_address() {
    $request_uri =
        isset( $_SERVER['REQUEST_URI'] ) ?
            filter_var( wp_unslash( $_SERVER['REQUEST_URI'] ), FILTER_SANITIZE_STRING ) :
            '';

    if ( url_to_postid( $request_uri ) ) {
        return '';
    }

    $path_parts = explode( '/', $request_uri );
    $part1      = isset( $path_parts[1] ) ? $path_parts[1] : '';

    if ( 'property' !== $part1 ) {
        return '';
    }

    $address = isset( $path_parts[2] ) ? $path_parts[2] : '';

    return str_replace( '-', ' ', $address );
}

/**
 * Filter page title.
 *
 * @param array $title_parts Title parts.
 *
 * @return mixed
 */
function theme_slug_filter_wp_title( $title_parts ) {
    global $parsed_address, $api_results, $wp_query;

    if ( ! $parsed_address ) {
        return $title_parts;
    }

    $title_parts['title'] = ucwords( $parsed_address ) . ' | AAA Real Estate';

    return $title_parts;
}

add_filter( 'document_title_parts', 'theme_slug_filter_wp_title' );

We catch 404 via pre_handle_404, make the API request and save results. Please note that it works when we have parsed address in the uri only.

Here is the tested code, you can use it as a mu-plugin:

<?php
/**
 * Filters whether to short-circuit default header status handling.
 *
 * Returning a non-false value from the filter will short-circuit the handling
 * and return early.
 *
 * @param bool     $preempt  Whether to short-circuit default header status handling. Default false.
 * @param WP_Query $wp_query WordPress Query object.
 *
 * @return bool
 * @since 4.5.0
 */
function pre_handle_404_filter( bool $preempt, WP_Query $wp_query ) {
    global $parsed_address, $api_results;

    $parsed_address = get_parsed_address();

    if ( ! $parsed_address ) {
        return $preempt;
    }

    // Make a request to  API, get status.
    $status      = true; // For debug purposes.
    $api_results = null;

    if ( $status ) {
        // Save results to $api_results.
        // Return true to avoid 404 processing by WP.
        return true;
    }

    return $preempt;
}

add_filter( 'pre_handle_404', 'pre_handle_404_filter', 10, 2 );

/**
 * Handle 404.
 */
function handle_404() {
    global $parsed_address, $api_results, $wp_query;

    if ( ! $parsed_address ) {
        return;
    }

    // Set 404 here to prevent bugs in get_header().
    $wp_query->set_404();

    get_header();
    echo 'API Results'; // Output $api_results here.
    get_footer();
    exit();
}

add_action( 'template_redirect', 'handle_404', - PHP_INT_MAX );

/**
 * Get parsed address from the request URI.
 * Make sure we have an address in the URI, otherwise return an empty string.
 *
 * @return string
 */
function get_parsed_address() {
    $request_uri =
        isset( $_SERVER['REQUEST_URI'] ) ?
            filter_var( wp_unslash( $_SERVER['REQUEST_URI'] ), FILTER_SANITIZE_STRING ) :
            '';

    if ( url_to_postid( $request_uri ) ) {
        return '';
    }

    $path_parts = explode( '/', $request_uri );
    $part1      = isset( $path_parts[1] ) ? $path_parts[1] : '';

    if ( 'property' !== $part1 ) {
        return '';
    }

    $address = isset( $path_parts[2] ) ? $path_parts[2] : '';

    return str_replace( '-', ' ', $address );
}

/**
 * Filter page title.
 *
 * @param array $title_parts Title parts.
 *
 * @return mixed
 */
function theme_slug_filter_wp_title( $title_parts ) {
    global $parsed_address, $api_results, $wp_query;

    if ( ! $parsed_address ) {
        return $title_parts;
    }

    $title_parts['title'] = ucwords( $parsed_address ) . ' | AAA Real Estate';

    return $title_parts;
}

add_filter( 'document_title_parts', 'theme_slug_filter_wp_title' );

We catch 404 via pre_handle_404, make the API request and save results. Please note that it works when we have parsed address in the URI only.

<?php
/**
 * Filters whether to short-circuit default header status handling.
 *
 * Returning a non-false value from the filter will short-circuit the handling
 * and return early.
 *
 * @param bool     $preempt  Whether to short-circuit default header status handling. Default false.
 * @param WP_Query $wp_query WordPress Query object.
 *
 * @return bool
 * @since 4.5.0
 */
function pre_handle_404_filter( bool $preempt, WP_Query $wp_query ) {
    global $parsed_address, $api_results;

    $parsed_address = get_parsed_address();

    if ( ! $parsed_address ) {
        return $preempt;
    }

    // Make request to  API, get status.
    $status      = true; // For debug purposes.
    $api_results = null;

    if ( $status ) {
        // Save results to $api_results.
        // Return true to avoid 404 processing by WP.
        return true;
    }

    return $preempt;
}

add_filter( 'pre_handle_404', 'pre_handle_404_filter', 10, 2 );

/**
 * Handle 404.
 */
function handle_404() {
    global $parsed_address, $api_results, $wp_query;

    if ( ! $parsed_address ) {
        return;
    }

    // Set 404 here to prevent bugs in get_header().
    $wp_query->set_404();

    get_header();
    echo 'API Results'; // Output $api_results here.
    get_footer();
    exit();
}

add_action( 'template_redirect', 'handle_404', - PHP_INT_MAX );

/**
 * Get parsed address from the request uri.
 * Make sure we have an address in the uri, otherwise return empty string.
 *
 * @return string
 */
function get_parsed_address() {
    $request_uri =
        isset( $_SERVER['REQUEST_URI'] ) ?
            filter_var( wp_unslash( $_SERVER['REQUEST_URI'] ), FILTER_SANITIZE_STRING ) :
            '';

    $path_parts = explode( '/', $request_uri );
    $part1      = isset( $path_parts[1] ) ? $path_parts[1] : '';

    if ( 'property' !== $part1 ) {
        return '';
    }

    $address = isset( $path_parts[2] ) ? $path_parts[2] : '';

    return str_replace( '-', ' ', $address );
}

/**
 * Filter page title.
 *
 * @param array $title_parts Title parts.
 *
 * @return mixed
 */
function theme_slug_filter_wp_title( $title_parts ) {
    global $parsed_address, $api_results, $wp_query;

    if ( ! $parsed_address ) {
        return $title_parts;
    }

    $title_parts['title'] = ucwords( $parsed_address ) . ' | AAA Real Estate';

    return $title_parts;
}

add_filter( 'document_title_parts', 'theme_slug_filter_wp_title' );

<?php
/**
 * Filters whether to short-circuit default header status handling.
 *
 * Returning a non-false value from the filter will short-circuit the handling
 * and return early.
 *
 * @param bool     $preempt  Whether to short-circuit default header status handling. Default false.
 * @param WP_Query $wp_query WordPress Query object.
 *
 * @return bool
 * @since 4.5.0
 */
function pre_handle_404_filter( bool $preempt, WP_Query $wp_query ) {
    global $parsed_address, $api_results;

    $parsed_address = get_parsed_address();

    if ( ! $parsed_address ) {
        return $preempt;
    }

    // Make request to  API, get status.
    $status      = true; // For debug purposes.
    $api_results = null;

    if ( $status ) {
        // Save results to $api_results.
        // Return true to avoid 404 processing by WP.
        return true;
    }

    return $preempt;
}

add_filter( 'pre_handle_404', 'pre_handle_404_filter', 10, 2 );

/**
 * Handle 404.
 */
function handle_404() {
    global $parsed_address, $api_results, $wp_query;

    if ( ! $parsed_address ) {
        return;
    }

    // Set 404 here to prevent bugs in get_header().
    $wp_query->set_404();

    get_header();
    echo 'API Results'; // Output $api_results here.
    get_footer();
    exit();
}

add_action( 'template_redirect', 'handle_404', - PHP_INT_MAX );

/**
 * Get parsed address from the request uri.
 * Make sure we have an address in the uri, otherwise return empty string.
 *
 * @return string
 */
function get_parsed_address() {
    $request_uri =
        isset( $_SERVER['REQUEST_URI'] ) ?
            filter_var( wp_unslash( $_SERVER['REQUEST_URI'] ), FILTER_SANITIZE_STRING ) :
            '';

    if ( url_to_postid( $request_uri ) ) {
        return '';
    }

    $path_parts = explode( '/', $request_uri );
    $part1      = isset( $path_parts[1] ) ? $path_parts[1] : '';

    if ( 'property' !== $part1 ) {
        return '';
    }

    $address = isset( $path_parts[2] ) ? $path_parts[2] : '';

    return str_replace( '-', ' ', $address );
}

/**
 * Filter page title.
 *
 * @param array $title_parts Title parts.
 *
 * @return mixed
 */
function theme_slug_filter_wp_title( $title_parts ) {
    global $parsed_address, $api_results, $wp_query;

    if ( ! $parsed_address ) {
        return $title_parts;
    }

    $title_parts['title'] = ucwords( $parsed_address ) . ' | AAA Real Estate';

    return $title_parts;
}

add_filter( 'document_title_parts', 'theme_slug_filter_wp_title' );

<?php
/**
 * Filters whether to short-circuit default header status handling.
 *
 * Returning a non-false value from the filter will short-circuit the handling
 * and return early.
 *
 * @param bool     $preempt  Whether to short-circuit default header status handling. Default false.
 * @param WP_Query $wp_query WordPress Query object.
 *
 * @return bool
 * @since 4.5.0
 */
function pre_handle_404_filter( bool $preempt, WP_Query $wp_query ) {
    global $parsed_address, $api_results;

    $parsed_address = get_parsed_address();

    if ( ! $parsed_address ) {
        return $preempt;
    }

    // Make request to  API, get status.
    $status      = true; // For debug purposes.
    $api_results = null;

    if ( $status ) {
        // Save results to $api_results.
        // Return true to avoid 404 processing by WP.
        return true;
    }

    return $preempt;
}

add_filter( 'pre_handle_404', 'pre_handle_404_filter', 10, 2 );

/**
 * Handle 404.
 */
function handle_404() {
    global $parsed_address, $api_results, $wp_query;

    if ( ! $parsed_address ) {
        return;
    }

    // Set 404 here to prevent bugs in get_header().
    $wp_query->set_404();

    get_header();
    echo 'API Results'; // Output $api_results here.
    get_footer();
    exit();
}

add_action( 'template_redirect', 'handle_404', - PHP_INT_MAX );

/**
 * Get parsed address from the request uri.
 * Make sure we have an address in the uri, otherwise return empty string.
 *
 * @return string
 */
function get_parsed_address() {
    $request_uri =
        isset( $_SERVER['REQUEST_URI'] ) ?
            filter_var( wp_unslash( $_SERVER['REQUEST_URI'] ), FILTER_SANITIZE_STRING ) :
            '';

    $path_parts = explode( '/', $request_uri );
    $address    = isset( $path_parts[2] ) ? $path_parts[2] : '';

    return str_replace( '-', ' ', $address );
}

<?php
/**
 * Filters whether to short-circuit default header status handling.
 *
 * Returning a non-false value from the filter will short-circuit the handling
 * and return early.
 *
 * @param bool     $preempt  Whether to short-circuit default header status handling. Default false.
 * @param WP_Query $wp_query WordPress Query object.
 *
 * @return bool
 * @since 4.5.0
 */
function pre_handle_404_filter( bool $preempt, WP_Query $wp_query ) {
    global $parsed_address, $api_results;

    $parsed_address = get_parsed_address();

    if ( ! $parsed_address ) {
        return $preempt;
    }

    // Make request to  API, get status.
    $status      = true; // For debug purposes.
    $api_results = null;

    if ( $status ) {
        // Save results to $api_results.
        // Return true to avoid 404 processing by WP.
        return true;
    }

    return $preempt;
}

add_filter( 'pre_handle_404', 'pre_handle_404_filter', 10, 2 );

/**
 * Handle 404.
 */
function handle_404() {
    global $parsed_address, $api_results, $wp_query;

    if ( ! $parsed_address ) {
        return;
    }

    // Set 404 here to prevent bugs in get_header().
    $wp_query->set_404();

    get_header();
    echo 'API Results'; // Output $api_results here.
    get_footer();
    exit();
}

add_action( 'template_redirect', 'handle_404', - PHP_INT_MAX );

/**
 * Get parsed address from the request uri.
 * Make sure we have an address in the uri, otherwise return empty string.
 *
 * @return string
 */
function get_parsed_address() {
    $request_uri =
        isset( $_SERVER['REQUEST_URI'] ) ?
            filter_var( wp_unslash( $_SERVER['REQUEST_URI'] ), FILTER_SANITIZE_STRING ) :
            '';

    $path_parts = explode( '/', $request_uri );
    $part1      = isset( $path_parts[1] ) ? $path_parts[1] : '';

    if ( 'property' !== $part1 ) {
        return '';
    }

    $address = isset( $path_parts[2] ) ? $path_parts[2] : '';

    return str_replace( '-', ' ', $address );
}

/**
 * Filter page title.
 *
 * @param array $title_parts Title parts.
 *
 * @return mixed
 */
function theme_slug_filter_wp_title( $title_parts ) {
    global $parsed_address, $api_results, $wp_query;

    if ( ! $parsed_address ) {
        return $title_parts;
    }

    $title_parts['title'] = ucwords( $parsed_address ) . ' | AAA Real Estate';

    return $title_parts;
}

add_filter( 'document_title_parts', 'theme_slug_filter_wp_title' );