Yes, I did investigate this further, but the issue is not within the search apparently, nor within the core W3TC plugin.
I see you have a custom made W3TC plugin as well, which I guess takes the localized inline script and puts it within a custom function. That is not correct, as it moves it out of the window scope and the ASP variable gets undefined.
If you are the one who made that, then make sure that the:
wp_localize_script('wd-asp-ajaxsearchpro', 'ASP', array(…
wp_localize_script('wd-asp-ajaxsearchpro', 'ajaxsearchpro', array(..
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