Hi!
It’s simply because the ajax handler for the add-to-cart buttons is not loaded for those buttons. You will have to load it manually with a custom script.
While customization is not part of the support, I may have found a solution for that for you. Put this code to the functions.php file in your active theme directory (from line 3 only):
https://gist.github.com/anag0/43ad32ec4a60ce50a27cbbcc312d9927
This will try to re-load the add to cart script to the ajax search elements as well.