Hi,
That error usually indicates, that either the server is not responding, or the ajax handler is not possible to access, due to an incorrect .htaccess configuration.
First, I suggest checking, if there is any malformed rule within the .htaccess file in the root directory. Then a malware filter/protection or firewall plugin can also disable access to the default ajax handler, I suggest checking that aswell.
You can also try enabling the custom ajax handler, it might be capable of bypassing the problem: https://i.imgur.com/FyjBDfJ.png
If nothing helps, you can add temporary FTP access to your ticket, and I will take a direct look to the code, to see if the request even reaches the search process.