SomethingClever:
The Feather is powered by USB and by a 3.7v Lipo battery when not plugged in. It has a built in battery charger so I don't have to worry about that.
Both power supplies are fine for a Pro Mini. It's got it's on-board regulator so will step down the voltage to the correct level.
The Pro Mini will only have the SD card reader attached and nothing else that draws power. Shouldn't it be able to handle that?
Probably can - it's the regulator that's the limiting factor.