Hi,
Okay, I think I see what you mean now. The problem is, that the search on the second site is specifically made that way, and is integrated into the theme. The elementor full screen search is also a similar thing, it is a completely different widget from the regular theme search, that cannot be simply replaced.
The closest possible feature in ajax search pro is the compact search bar. You could maybe try using that on the mobile menu as a shortcode. You can even use a fixed layout for it, so it bascially floats on the sidebar, like on our demo.