Perfectly reasonable questions. It's mentioned in one of my stories where this sort of thing happens that such a pred has little choice but to just relieve itself right where it sits until it's digested and passed enough to waddle away. This isn't something I draw, but it happens.

As for the badger, once again in stories it is mentioned that a badger who gets this full in its burrow abandons that bit of burrow afterward due to it being full of poop. There's even one story where investigators poke through the poop looking for clues, though it is largely off camera.