Hi,
Thank you very much for the details, it helps a lot!
The issue is, that on the desktop version the built-in theme live search box is visible – which acts on it’s own – and on the mobile it’s the Ajax Search Pro search input giving it’s own configured results. So they are basically two different searches.
You can actually keep it this way, but make sure to configure Ajax Search Pro so that it returns the post types you want (I suspect Products only).