I have moved your reply to a new ticket, it's more convenient this way.
I checked the website and the problem is with the cloning. All of the ajaxsearchpro instances are using a unique id. The menu clones all of it's contents including the search, so there are two elements with the same id's on one page. Two elements with the same id's on the same page cannot exist, it's against the w3c standards.
Besides that the client side cloning faces another issue, that even if there was no ID on the search element, the initialization event was fired when the page was loaded, and that step cannot be repeated. (it's the jQuery(document).ready(…))
A way to solve this is probably by editing the search id when it's cloned (or after cloning), I'm not sure if it's possible, and then make a new initialisation event for that new search instance.
I can take a look at that if you can send me temporary ftp/admin access, but I cannot promise that I can solve this (it might be impossible). Please open a new ticket with the login details and I will take a look at your site.Best,
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