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This is what I do, although I have a broody pen in my coop that's about 5' x 6', simply because it's easier than trying to move the broody, and there are no integration problems. If you put her in a dog crate she can't get far from the nest to poop and can't really get any exercise. Mine will go outside, forage, dust bathe, come in and fly up to the roost, clean her beak on it -- about 15 or 20 minutes of normal chicken life. I just mark the eggs with a Sharpie and check under her once a day.
Recently when I tried to move a broody she stopped being broody. Also had one fly/climb over a 6' fence, leaving her eggs behind, to get back to the nest where she'd started setting on eggs.
You'd be taking a lot of space away from your flock if you made a comfortable broody pen in there, especially if they are staying indoors in this weather. I would not do it.