Some people mix sizes like that and some find it doesn't work out. A lot of people have a separate "bantam coop." If your red crosses are anything like the production reds I've had, which are crosses, I definitely wouldn't try it as those hens have been pretty aggressive.
I would not add a lone bantam to an existing flock of large birds, because a flock virtually always gives the new comer a hard time, and if the new one is weak or small, it could get really violent if not lethal. But I do mix bantams and big ones when I raise them all together from hatch. My rule of thumb is that the bantam has to have at least 1 friend her own size, and usually more. Right now I have 8 bantams and 7 large birds cohabitating quite peacefully.