Maybe dry it and add to their nest boxes. I do this sometimes, a little lavender too from time to timeMint will take over like a weed if you let it. Some of them will string out of the original pot and spread where they touch the ground. Not necessarily a bad thing.
I heard from a few people, which I need to check if it's true or not, they are experimenting with mint in beehives to keep the hive beetles out. If this is true, I wonder if it'd keep certain pests away from the girls too, if you can keep them out of it. If not, at least their poop might smell a bit nicer
Aaron