> 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.
By the way, for the e-ink project I just linked in the other comment I ended up discovering the RV-3028-C7 RTC module. Idle power draw 45 nA at 3.3V instead of 110 μA for the DS3231 for similar functionality and ±1 ppm accuracy with an internally sealed oscillator. Awesome for battery-powered stuff and cheaper at low qty on Digikey.
I hope more of those little DIY/maker breakout boards adopt it. Pimoroni sells one. The one in my project is a free sample dev board that Micro Crystal sent me on request - which I guess is working out for them considering I just shared their product with you all.
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.