Well, it is outside of Ajax Search Pro, but I will try to help. The only issue with the embedded search is that it’s not having the “post_type=product” argument, which is the trigger for WooCommerce to capture and display it’s product results page.
Maybe you can set up a .htaccess redirection, or a programmatical redirection so the post type argument is added.
Even simpler might be to add an extra hidden input to the modal search with the post type argument, so it’s autoamtically captured by the woocommerce results page. That might be the simplest possible solution in my opinion.