Well, that took all of ten minutes to do. That was super easy! There was already a "C" terminal on the heater's main control board, easy to reach, easy to see, and there was an unused blue wire in the wire bundle, just like the guy said. Strip both ends of the wire and connect to the "C" terminals at either end, and the hack is done. Nest tells me it's got a 24v common wire detected now.

I have no way of knowing whether that's gonna *fix* my problem or not, though. I'd rather still find out what my options are for finding a replacement battery for this thing. Anyone know?