Hardware version 1.3.1 works perfectly so far as I can tell. If I had a complaint it's that the audio volume is a little low. It turns out it's the fault of the low-pass cap on the amplifier feedback. Changing it form 2200 pF to 1000 pF fixed the problem. I did also bump up the feedback resistor from 20 kΩ to 56 kΩ. That boosted the volume a little bit, which will probably be about right for when the board is installed in the case. Also, increasing the anode pull-down resistors from 10 kΩ to 100 kΩ didn't result in the ghosting coming back. On the scope there's just a little bit of a tail still at turn-off, but it's just a little less wasteful now when the anode is switched on.
The power problems are all solved. The gate of the power MOSFET is effectively right at battery voltage, and I can't find a mA of leakage from the batteries. As before, the whole system seems to operate just fine at 2.3 volts and starts to flake out at 2.2 volts (from my bench supply). Of course at that voltage it's drawing something like a quarter amp, so if the batteries are that tired, they probably won't keep up in any event.