Hi,
I assume the full screen search is custom coded in some way, or it may use the output from the theme search.php file.
I think the best solution is to turn the automatic search replacement off, and instead use the ajax search pro shortcode directly in the Woodmart header, instead of the search module for desktop.
I believe their header builder has shortcode support via text/HTML modules and you can still keep their search module on the mobile view, only replacing the desktop search.