So I got the latest copy of the code, but could not duplicate the problem. Next I went and retrieved an exact copy of the code that CM used for their official build. We actually have configuration management practices that let me trace the version of every file used in the build. But even with this version of code, I could not make the problem happen in Debug or Release mode.
The CM Team recommended that I let them try to do a build again, in case this was a one-time fluke. No such luck. The problem persisted even in the new build. So I went to the build machine and ran the application built there from Visual Studio. Could not make the problem happen. At this point I knew we had a very strange problem on our hand.
So I went through our whole application suite, trying each executable and DLL one at a time. I still could not make the problem happen after copying all our target files to the development folder. So for kicks I tried copying some of the system files we deploy during install. And this is where I stumbled upon the problem. Turns out the MFC DLL we ship was not getting properly installed into the Windows system directory.
Since I was in a hurry, I just changed our install code to copy a local copy of the MFC DLLs to our application directory. This works but is not elegant. Unfortunately it may have to stay this way.