I think cross breeding will be problem even if harems (hens) and master (rooster) are of same breed. Harem masters do not seem to copulate with hens outside their harem very often but the satellites (roosters w/out harems) do jump hens whenever they can. Most chicks should still be out of harem master.
When I was younger we kept American games on walks on our farms and farms of neighbors. On our farms in particular, we had enough outbuildings and feedlots to enable multiple harems (sub-flocks) to setup nearly allopatric (non-overlapping) ranges. Each subflock would have a feedlot (hog or cattle), watering hole (livestock pond or large puddle), and roost. Some sub-flocks shared same building as roost but one group would have one manger head and another sub-flock would have another and in big barn a third group would roost on hay rope in loft. I have had success setting up sub-flocks without out buildings by raising groups of chicks together and providing each sub-flock with a roost and feeding stations. Some overlap has occured at feeding stations, sometimes with conflict, sometimes not, but sub-flocks have remained intact and discrete. Also keeping adults together for several weeks causes them to bond. Acreage and more investment in feeding stations I think enables. Too many birds and not enough habitat does not work in favor of sub-flock stability. Whenever sub-flocks broke down, a stag (satellite rooster) was involved and it often resulted in multiple harem masters getting involved. With American games system not stable as a result. Take home point to promote stability, pen up roosters that are not harem masters. If breeds that are allowed to co-mingle can be distinguished from their hybrids, then such as system will work for you. You will be addressing crosses as part of culling process but labor may in end be less.
