UPDATE!
I followed the troubleshooting steps in the factory repair manual and everything kept coming back to bad alternator (or at least a bad diode in the alternator). Finally, I sucked it up and took it back. I convinced the parts counter guy that the one he just sold me was defective right off the shelf and I went home with another new one... happy day, that fixed it!
This shouldn't have been such a painful thing to put right - I've never had a "dead on delivery" part before.