A.S.
In my experience, even in a Christian school the behaviors that you are seeing are part of being socialized primarily by your peers (instead of your parents).
Believe it or not - where I have seen the strongest "anti-clique" mentality has been in our homeschool groups. I think it's because the kids spend significantly more time with adults who care about them, and when they do get together with other kids they are not as desperate for attention, dominance, etc. (and thus more likely to pick on one another). Our homeschoolers have been the strongest demonstrators of their faith, too (not that all homeschoolers are perfect - not making that argument).
Prior to homeschooling, we were in a couple of different private schools for a good number of years - and in every school I saw the clique situation get worse with age (and it really seemed to peak in 7th and 8th grade).
I applaud your initiative to nip it in the bud early - I'm just not sure it can be done due to the innate structural inadequacies of the way we educate children today. I hope you can do it though. Good luck.