Hmm. I tried running as admin and still got the "Registration data is invalid" dialog box. The new serial number showed up on the splash screen, though.
I tried a couple of other things, too (running under XP compatability mode and setting the privs for tmg.exe for both Administrator and me to "Full Control"). No joy.
So I tried uninstalling, and for good measure scrubbed the registry, then reinstalled (I'm using the from-scratch tmgsetupv6.exe). No change.
UPDATE: I finally got it, after another fruitless reinstall.
I had to type in the serial number instead of pasting it.
The problem is/was that this is on a Mac, using the Parallels VM. The email is naturally a Mac program, and I was copying and pasting the serial number from the email. There must be an encoding issue between the two (IIRC, Mac OSX uses UTF32 and MS-Win uses UTF16, for what that's worth), and the pasted result, although it displayed correctly, wasn't the correct value for the hashing function.