If the error check page shows no error, that’s a good thing so far.
This issue was usually related to a caching incompatibility, or if the database query is interrupted when the plugin is installed. That’s why I created the upgrade steps in the documentation, to avoid such issues. I remember one time someone had something similar, but reinstalling helped them.
This sounds like something else, but I can’t confirm until I can see it or do some kind of debugging. (checking for errors, printing the database query and comparing it to the actual configuration, etc..)
Could you at least provide temporary administrator access and a link to the site? I can check the configuration at least.
I can also try to make changes via the plugin editor in wordpress, but on the downside if I make a syntax error by mistake your site will go down with a white page most likely, and the only way to revert is to correct the mistake via sftp or rename the ajax-searh-pro plugin folder (or delete it completely).
If you like my products, don't forget to rate them on codecanyon :)