I would have made the same diagnosis as the Apple Store, but I still don't like not having my computer for another fucking week. After it came back with a fresh logic board, I ran it for about 3 hours on the battery. I plugged it in at home, did some more work for about 45 minutes, and as I was connecting to my computer to transfer Lee's files, this time it at least gave a kernel panic screen. Subsequently it wouldn't boot (but, unlike before, would give a boot chime).
So the powerbook is back off to the Depot, but now we know that there is a problem specifically with DC power. I'll also exchange my adapter, just in case.