While I do understand that, please also note that I am not using a menu plugin on mobile.
This is stock standard WordPress submenus?
If you say that there is no solution for this, then your documentation/sales page needs to specifically say something like:
“Please note that you cannot use this search with the default mobile submenu on WordPress”.
I am not happy with this as a resolution, and implore you to figure out something for both myself and future customers. This has basically enough for the client to remove this plugin and look elsewhere.
Thanks