Execute jQuery with custom event listener after successfully add an item to the cart

From a cursory overview, the cross-sells are added via the same ajax add to cart script as regular shop loop archive products and I don’t see a way to distinguish between when it’s a cross-sell and when it’s not. You could run a script conditionally only on the cart page, and then listen to the added_to_cart trigger, which might then only be happening for cross sells (assuming you don’t have other ajax add to cart buttons on the cart page)

Or it appears that the add to cart button is passed as a variable to the added_to_cart trigger. You may be able to do some DOM manipulation there to see if it was part of the cross sells div. Totally wild guess:

$( document.body ).on( 'added_to_cart', function( event, fragments, cart_hash, $button ){
    if( $button.closest( 'div.cross-sells' ) ) {
        // was a cross-sell added to the cart
    }
});