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If it's a Permethrin based dust labeled for use in poultry, then it should be fine - you would want to apply dust your broody under the wings, around the vent and along her back. Do it outside of the box, so she can shake off excess. I do understand your concern, but you need rid of the mites.

Yes, I would treat the broodies, and if possible the nesting box. Can you safely remove her bedding as well? When she hatches the eggs, monitor the babies too, if you see any signs of them getting mites on them, then give them a light dusting as well. Mites will suck the life out of them...