Add custom button to the changeset status in the Customizer

Unfortunately there is no official hook for customizer actions yet. Since the customizer actions are predefined by javascript, you can use the The Customizer JavaScript API to add new actions via the push function.

PHP function for the index.php/functions.php:

    // Include scripts in customizer.
    add_action( 'customize_controls_enqueue_scripts', 'script' );
        
    function script() {
wp_enqueue_script( 'custom-customize-controls', plugin_dir_url( __FILE__ ) . 'customizer.js', [], null, true );
    }

and the framework customizer.js file…

;(function (api) {
    api.bind('ready', function () {
        api.settings.changeset.statusChoices.push({"status":"action","label":"Action"});
    });
})(wp.customize);

Does this option work for you?