James,
I don't think you're going to need to recompile. For one, Redhat's "automatic configuaration GUI" doesn't use source files.... uses RPM's (basically pre-compiled binaries). Thus, you probably don't have the source readily available and would need to download it. That's not a big problem though🙂
More to the point, AFAIK, the RPM's are compiled using apxs and the --with-mysql option is default (you have to manually specify that you don't want MySQL support). Your best and easiest bet is to add one or more RPM ftp servers into the sources list of the PRM Manager (don't ask me how.... I don't use Redhat) and run a search for php-mysql. It'll turn up a bunch of hits (probably on the installation cd's as well) for the php-mysql rpm, just select it and install it.
In answer to your other question, the make options you've given in your post are valid and won't cause the make to barf and die (assuming you've got MySQL and APXS installed on your box). Personally I find it's useful to place the "make clean" command before the ./configure options. It ensures you're running a fresh install and not using an old cached config script.
-geoff