Hi,
The problem is that the menu actually makes a DOM clone of the search bar in mobile view, so the same searcb bar is ‘copied’, creating invalid HTML and loss of attached event handlers. This is a very bad practice, and it cannot be resolved with any custom code or option – as the issue is caused by the html cloning itself.
The only possible resolution is to use the search bar in a different position instead. Some menus are simply not made to hold interactive (javascript powered) elements. A simple search form without any features would work, but anything that uses javascript will most likely fail in that position.
I hope designers will stop using this copy/clone method in menus once and for all, and use CSS media queries instead for much better compatibilty. It is already very rare, but still used.