I have an update on this. I found that using the Ele Custom Skin plugin (https://wordpress.org/plugins/ele-custom-skin/) and enabling the option “Live load the results on the results page?” in Ajax Search Pro, works well. The Ele Custom Skin plugin is an enhancement of the standard Elementor Pro Posts-widget, allowing it to use a custom template for showing the results.
You can try it here: https://worldofsuperyachts.com/engines/ (try typing “cat”).
However, the Dynamic Content for Elementor plugin, is a far better plugin, and I would still prefer the integration with them. But the above is a nice workaround for now.