Here's how I would go about doing this:

First create a file with a hash for the file name within your theme directory:

3858f62230ac3c915f300c664312c63f.php

Within that file would be something like this:

//Use the file name as an API key of sorts
$file = explode('.', basename(__FILE__));
$key = $file[0];

//Trigger the WP function - corresponds to the foo_w3tc_flush_cron() function
define('FOO_W3TC_FLUSH', true);
define('FOO_W3TC_FLUSH_KEY', $key);

//Set headers
Header('Cache-Control: no-cache');
Header('Pragma: no-cache');

//Set W3TC Constants
define('DONOTMINIFY', true);
define('DONOTCACHEDB', true);
define('DONOTCACHEPAGE', true);

//Set WP Constants
define('DOING_AJAX', true);

//Load WP
if(file_exists($_SERVER['DOCUMENT_ROOT'].'/wp-load.php'))
    require_once($_SERVER['DOCUMENT_ROOT'].'/wp-load.php');
die();

Add a function to your functions.php file similar to the following:

$api_key_plaintext = 'foobar';
define('FOO_W3TC_FLUSH_API_KEY', md5($api_key_plaintext));

add_action('init', 'foo_w3tc_flush_cron', 1);
function foo_w3tc_flush_cron(){
    $update = FOO_W3TC_FLUSH_UPDATE;
    $key = FOO_W3TC_FLUSH_UPDATE_KEY;
    
    if($update && $key == FOO_W3TC_FLUSH_API_KEY){
        if (function_exists('w3tc_pgcache_flush')){
            if(w3tc_pgcache_flush())
                wp_mail('[email protected]', 'Success!', 'Hello, this is an automatically scheduled email from WordPress.');
            else
                wp_mail('[email protected]', 'Failure', 'Hello, this is an automatically scheduled email from WordPress.');
        }
        echo "Foo W3TC Cache Message"; //For logger
        die(); //We don't need the rest of WP to lead
    }

}

Finally, I would add the following crontab to run at 11:59pm (your server time) and include a file path to your home directory to log results:

59 23 * * * curl --header 'Cache-Control: max-age=0' http://your-domain.com/wp-content/themes/your-theme/3858f62230ac3c915f300c664312c63f.php >> /home/myuser/w3tc-flush.log

Hope this helps out!

Here's how I would go about doing this:

First create a file with a hash for the file name within your theme directory - this one is md5('foobar'):

3858f62230ac3c915f300c664312c63f.php

Within that file would be something like this:

//Use the file name as an API key of sorts
$file = explode('.', basename(__FILE__));
$key = $file[0];

//Trigger the WP function - corresponds to the foo_w3tc_flush_cron() function
define('FOO_W3TC_FLUSH', true);
define('FOO_W3TC_FLUSH_KEY', $key);

//Set headers
Header('Cache-Control: no-cache');
Header('Pragma: no-cache');

//Set W3TC Constants
define('DONOTMINIFY', true);
define('DONOTCACHEDB', true);
define('DONOTCACHEPAGE', true);

//Set WP Constants
define('DOING_AJAX', true);

//Load WP
if(file_exists($_SERVER['DOCUMENT_ROOT'].'/wp-load.php'))
    require_once($_SERVER['DOCUMENT_ROOT'].'/wp-load.php');
die();

Add a function to your functions.php file similar to the following:

$api_key_plaintext = 'foobar';
define('FOO_W3TC_FLUSH_API_KEY', md5($api_key_plaintext));

add_action('plugins_loaded', 'foo_w3tc_flush_cron', 1);
function foo_w3tc_flush_cron(){
    $update = FOO_W3TC_FLUSH_UPDATE;
    $key = FOO_W3TC_FLUSH_UPDATE_KEY;
    
    if($update && $key == FOO_W3TC_FLUSH_API_KEY){
        if (function_exists('w3tc_pgcache_flush')){
            if(w3tc_pgcache_flush())
                wp_mail('[email protected]', 'Success!', 'Hello, this is an automatically scheduled email from WordPress.');
            else
                wp_mail('[email protected]', 'Failure', 'Hello, this is an automatically scheduled email from WordPress.');
        }
        echo "Foo W3TC Cache Message"; //For logger
        die(); //We don't need the rest of WP to load
    }

}

Finally, I would add the following crontab to run at 11:59pm (your server time) and include a file path to your home directory to log results:

59 23 * * * curl --header 'Cache-Control: max-age=0' http://your-domain.com/wp-content/themes/your-theme/3858f62230ac3c915f300c664312c63f.php >> /home/myuser/w3tc-flush.log

Hope this helps out!