Ok, so I did some more testing. There was a strong chance that, as it's a hand-made board, something could have got disconnected/shorted/broken/whatever, but when it was last in use, it worked fine.
So I loaded firmware 20140809 into it and configured it with an old version of MDM. The problem has disappeared entirely. It's been on for the past 20 minutes without a single false trigger, so I suspect some kind of bug has crept in since then. I also suspect that there aren't many atmega644/1284 users left, and the ones there are probably haven't updated their firmware in a while. So unless anyone else complains about the problem, I'm fine using this older firmware since it does what I need.