Based on that error it seems that the PHP mysql extension is not enabled. You should be able to enable it from the php.ini file.
(and sorry, I didn't receive a notification of your reply, hence my slow response)
The good news is that I was able to enable mysqli using php.ini and Perch seems to work as expected even after updating .htaccess with the suPHP_ConfigPath command.
The not-so-good news is that I've not yet gotten the suPHP_ConfigPath to work as expected, so even after I increase the PHP settings in php.ini Perch does not "see" them. But perhaps that is a thing for another thread, if needed.
Thanks for your help!
Edit: of course, right after I posted this, I got it working. Typical.