Hi,
Thank you for your kind words and all the details, it helps a lot.
Usually I recommend an option or a custom code where I can, but this one I’m afraid is not possible to change at the moment. However the interface is in the code and there is an option to disable this for the regular live results, all I need to do is make a few changes to make it work with the Elementor live results feature as well.
I will make sure to implement this in the upcoming release, so by default the URLs will not change, unless set in the back-end. In the meantime, if you want, you can add temporary back-end and FTP access, and I can try to make a minor change to the plugin scripts to not to update the URLs.