Any question answered...

Don't have the foggiest about electronic shifting. But I guess most likely is something in the circuit that drives the derailleur servos? Although I would be surprised if it worked at all if something like that was gone. A good test would be to borrow someone elses battery and see if it behaves the same. Or wire up a little circuit with known resistance, run it from a fully charged battery and see if it lasts as long as it should.

It's a made rather than bought battery, I could always put another one together though. I'm just a little cautious as the chap who made the current one said that soldering onto Li-Ion batteries could be "life changing", I presume in a bad way.