Hello and welcome to BYC! Glad to have you here!
I personally don't think it would be an issue to add a Brahma so long as they are introduced properly. Whether or not the mud is an issue depends on exactly how bad it is. I have thick clay soil in my area, and even my birds (who have no feathers on their legs) can get very weighed down if they walk in the wrong spot for too long. They have wood chips and old leaves around their run to prevent that from happening, so that might be an option for you if you're concerned about mud. Best of luck!