Well, the thing about Rhode-Island Reds is that their personalities are very hit-or-miss. Some will be really sweet towards other birds, others will be really mean, to the point of bloodshed.
My parents raised RIRs for almost 11 years, and they were some very feisty and bossy birds. They never got along with other breeds, even if the breed was bigger than they were! Even my nicest, most docile RIR hen, Jesse, did not like my other breeds we tried to introduce.
Honestly, if it were me, I'd say NO to putting the bantam cochin with the RIRs, but since your RIRs seem to be okay and pretty even-tempered, you could give it a shot. I'd just watch for a few days and see how it goes, and so long as no one's killing each other, all should be golden.
I also read that chickens have a strict hierarchy among them, and adding a new member throws everything out of whack, and they have to 'reassert' themselves into the order again.
(Now, these ideas I have listed here I never had a chance to try on my flock, because I didn't know nearly as much about chickens as I do now. Research has done wonders!)
-Some people recommend putting the new member in a cage in the coop for awhile, to let the new kid and the other gals get used to each other from a distance.
-Also, once they're in the cage for awhile, you should probably introduce the new kid at night while the other chickens are roosting, to give the others the idea that the new member is "supposed to be there."
-Another good idea would be to give the flock plenty of other distracting activities, like hanging some leafy greens up high just barely within reach, to get the birds' attention and give them some exercise, too. Quite a bit of chicken-picking comes from them just being bored and having nothing better to do than pick on each other.
Chickens will be chickens.
Hope all this helps, and good luck!
