In this type of question I usually ask about your facilities and give you some options. I don't think that is necessary in your case, you've answered those questions well enough. You need to relocate that broody hen. Fix her a space with a nest, food, and water and not much else. She needs a little room to go poop but not a lot. She should know to not go poop in her nest but that instinct does not extend to her food and water. You will probably be cleaning them out so give your self decent access.
You need a pen that you can lock her into until she hatches, one that she cannot get out of and no other hen can get in. I'd move her at night and move her eggs with her. The biggest risk is that she will break from being broody. Most hens will accept the move to a new nest but occasionally you get one that won't. I think it helps them accept the move if the new nest is sort of dark, or at least not bright.
Good luck. Let us know how it goes.