I need to get some pictures put up of the new board in the new case running the new software (all of which is up on GitLab), but it's up and running.  When discharging Li-ion cells, the load resistors get just as toasty as they did before: hot, but not nearly so hot that solder would start to melt.  (I've seen that happen, on a computer motherboard...while the computer was running.)  I've put up a picture of the board (with display plugged in) in the meantime.

The updated software uses the Adafruit_GFX library instead of u8g2, which I couldn't get running on the ATmega328PB.  Screen updates seem to be slower, but the only effect I can see is that it takes longer to begin cycling through battery positions when you hold down the button.  Switching between libraries was fairly easy, and the core routines work the same as they did before.