I decided I had nothing to loose so I cracked open the old unit and found a battery. If course, it wasn't any kind of normal battery. That would be too easy. Fierce googling ensued and eventually I found something as close as I could and trusted to fate.
The battery arrived today and damn if it wasn't a perfect fit. I even managed to find all the screws to put it back together again. Total cost? $19. There is indeed something deeply satisfying about fixing something yourself. It would have been easier to just buy the new inflater or a new computer system but this was I saved a few bucks and learned a little bit in the process.