> small CR2032 coin cells (which have ~10 to ~100 ohms of internal resistance). A ~100mA draw on the charge inductor isn't out of the question (at 10-ohms, that's a voltage drop of 1V, which probably browns-out the RP2040).

n=1, but I'm using a single CR2032 with a Badger 2040 (same as OP) and a DS3231 in parallel and it works just fine down to ~2.8 volts.