How to perform a frontend HTTP call with AJAX when plugin save the new settings?

With that code in place, all you should need to do is poke the page with wp_remote_get() as your backend page loads, and before you need to retrieve the option value. By doing this, you avoid the possibility that no visiter has visited the page. Your request will be that visitor.

The function is fairly self-explanatory:

$response = wp_remote_get( $url, $args )