Trying to run Javascript on AJAX call

Going by your method I would suggest hooking into wp_footer as well, so that your script would be near the end of the body, that way it will be loaded after all DOM was loaded.

add_action('woocommerce_ajax_added_to_cart', 'set_button_classes');
function set_button_classes () {
    add_action('wp_footer', function () {
        echo "<script>document.getElementByClassName('added_to_cart wc-forward').classList.add('ct-cart-item', 'ct-offcanvas-trigger');</script>";
    });
}

You could also close the php tag and reopen it in the wp_footer anonymos function to make it easier to understand/see the scrip code

add_action('woocommerce_ajax_added_to_cart', 'set_button_classes');
function set_button_classes () {
    add_action('wp_footer', function () {
        ?>
        <script>
            document.getElementByClassName('added_to_cart wc-forward').classList.add('ct-cart-item', 'ct-offcanvas-trigger');
        </script>
        <?php
    });
}