I figured something out based on what I found at this [post][1].

Basically I need a plugin that listens for when the REST response is about to go out. The plugin code would be similar to the following:

    function ag_filter_post_json($response, $post, $context) {
        $tags = wp_get_post_tags($post->ID);
        $response->data['tag_names'] = [];
    
        foreach ($tags as $tag) {
            $response->data['tag_names'][] = $tag->name;
        }
    
        return $response;
    }
    
    add_filter( 'rest_prepare_post', 'ag_filter_post_json', 10, 3 );


It adds the tag names as a new property called `tag_names`. The rest of the heavy lifting is done by the [`wp_get_post_tags`][2] function.


  [1]: https://www.haselt.com/blog/wp-rest-api-modifying-the-json-response
  [2]: https://codex.wordpress.org/Function_Reference/wp_get_post_tags