Richard Brooksby created a Perforce integration tool for Bugzilla and TeamTrack. You can find the main project page at http://www.ravenbrook.com/project/p4dti. "p4dti" is now an officially supported product from Perforce, and you can find the "Perforce Public Depot" p4dti page at http://public.perforce.com/public/perforce/p4dti/index.html.
Integration of Perforce with Bugzilla, once patches are applied, is fairly seamless. However, p4dti is a patch against the Bugzilla 2.10 release, not the current 2.12 release. I anticipate patches for 2.12 will be out shortly. Check the project page regularly for updates, or take the given patches and patch it manually. p4dti is designed to support multiple defect trackers, and maintains its own documentation for it. Please consult the pages linked above for further information.
Right now, there is no way to synchronize the Bug ID and the Perforce Transaction Number, or to change the Bug ID to read (PRODUCT).bugID unless you hack it in. Additionally, if you have synchronization problems, the easiest way to avoid them is to only put the bug information, comments, etc. into Bugzilla, and not into the Perforce change records. They will link anyway; merely reference the bug ID fixed in your change description, and put a comment into Bugzilla giving the change ID that fixed the Bugzilla bug. It's a process issue, not a technology question.