I’ve already contacted the theme support and haven’t received any solution.
That is really disappointing, it should be a super simple code or widget replacement, not sure why they don’t care. I have seen at least a 1000 other search bars like that in theme headers replaced without any particular issues.
Can you share like a copy of the theme or temporary sftp and back-end details to a test server where I can check out and debug the issue directly? Even if that search is hard-coded to the theme, it should be a 30 second replacement within the header file through a child theme.