So to everyone who's posted with this annoying problem, I just want to say that it appears to be fixed in an upcoming release.
Without going into any further detais about the release itself, so as not to incur the wrath of Apple, I submitted this as a bug to the original 10.8 beta, and to the beta of an upcoming build. A recent build release seems to have fixed it.
I am still using Thursby pkard as my middleware (software to read the CAC), and I can't verify that other middleware products won't continue to have problems. But with pkard at least, I can log into CAC websites, or use my CAC to sign email, and my preference panes can still be locked and unlocked without any problem.
I would be interested in hearing back from anyone who still has a problem after the 10.8.1 release, just to see if the problem is fixed across all middleware. It should be, as the problem was in the OS.