The second prototype build was more or less successful. The one problem remaining to solve is the impact the audio amplifier has on the 3.3 volt power rail.
47 µH of inductance followed by 47 µF of capacitance feeding the amp didn't fix it. I'm not sure where to go from here.
One minor detail is that the controller's brown-out detector needs to be configured or else the soft power system becomes unreliable. As the system power drops to the point where the controller is unreliable, it can result in spurious resets, and a reset will cause the power pin to be pulled low (by software). The brownout system has a threshold higher than the chip's minimum and has hysteresis. It's no big deal - just an extra step for programming to set the proper fuses.
Well, I have a couple of choices as to what to try next to solve the power problem. One tack is to power the audio amp directly from the battery. This means that it won't be getting a well-regulated supply voltage, and the audio quality will suffer as the batteries drain. But in testing with the prototype, doing it that way does make the system reliable. The other option is to try to go with a beefier boost converter. The nominee is the TPS613221A. It can supposedly provide a bit more current, particularly when configured with an external "helper" Schottky diode. I think I'm going to put in orders in to OSHPark for both options, build them both, and see which works best.