So, guess what... as of this morning, they are both fully charged and with the charger plugged in, neither is showing this effect!!!
I can only suppose that it was a combination of running the ATMega328 at 3.6 volts and the fact that the battery being less than fully charged dragging down the charging controller output below a point where the processor could run reliably. Since they are now both immune to this effect, I'll just keep an eye on them after I've discharged the batteries a bit to see if it returns.
I took out my old Weller WM-120 12-watt iron with the tiniest tip I have and reflowed the solder on the dropping resistor for the STOP LED, and now the LED is stable (had been flickering with any flex of the board). Another problem solved.
My next project will be hacking in a connection to the serial port on one of them. The challenges there are threefold:
1. The TX/RX pins are right between INT0 and RESET so I have to be super careful soldering tiny wires to them.
2. The ATMega328 spec says the max voltage on the signal pins is Vcc+0.5 volts, so I'll have to work out something to guard against 5v in.
3. This looks to NOT be a multi-layer board, so I'm hoping it will be safe to drill holes for a 4-pin header clear of the traces I can see.