Clock Radio
Battery Protection Bypassing
The resistor I thought was important in the BMS is irrelevant (at least as far as I can tell). Looking at documentation for the compatible and far more common DW01, the part that measures the current is the voltage across the MOSFETs. So it depends on the Rds(ON) of your MOSFETs.
The version of the BMS chip I got has a lower overcurrent threshold (there are many different parameter variations you can buy and this was one that was available). When this starts up, it has to fill these enormous power capacitors. That’s probably pulling an amp or two for a few milliseconds.
So I tried taking the MOSFETs (and thus the current sensing) out of the picture by shorting across them. With that bypassed, it starts up fine. Stable 5 and 3.3V.
Though it actually took a couple of seconds to reach a voltage where a red LED would light up, so that RC time constant to fill the capacitors was probably a bit longer than I guessed.