I'd let her move into the burrow with me, only I'm not sure she'd fit - rabbit holes aren't very big. And when it rains, the burrow isn't much drier than the box, anyway.
But it's rather a moot point. Eenie spends so much time in the dungeon these days, I'm not sure she even knows where the box is. 
But would an invisible fence interfere with your collar, my queen? Or do you no longer wear it?