I can report that disassembly of a battery and desoldering-resoldering of the microchip from the leads connecting it to the cells does appear to clear up the microchip controller's confusion. Like you'd expect, I might add.

An older but still new-ish battery than the one mentioned in the post above was treated to this process this morning. It had failed about a year ago. No cells are touched in the soldering, so doing this is about as safe as fiddling with a battery with a soldering iron gets. It involves cracking the plastic seam and opening the battery (I tapped gently on the seams with a broad chisel -- works a treat, though care must be taken, as all you want to do is break the glueline, not drive the chisel into the battery!), disconnection of the leads on the microchip controller (I did all four for the sake of thoroughness, but presumably disconnection of + and - would do the trick too), tapping the old foot for a few minutes to make sure all was cleared, and then reassembly. Took me about half an hour, all in.

The battery was previously at 0% and would not charge. All recommended power management measures etc had been taken, repeatedly, to no effect. After the treatment described, however, the battery immediately began to charge up on insertion in the machine and behaved normally. It is now giving me a four hour charge. So the battery's having its head cleared seems to have done the trick.

Of course, the question is whether this is a fluke or would work more generally, and whether it will last. I don't think it a fluke, since there was method in the madness, but I will report back in a week or two on how the battery behaves itself following cycling a few times, and I will then give the other battery the same treatment to see the effect on it.

Evidently the firmware in these things is a delicate flower.