Kids don't bully because they're poor. They bully because the other kid is different (e.g. jocks vs geeks), they bully to gain peer points, and for many other reasons, that are orthogonal to public vs privaye, or poor vs rich kid.

As for taking to others bad home experiences, like mimicking an abusive parent, that can happen to rich as well as to poor kids...

Yes, of course bullying will happen in better off schools as well. I just think there will be less of it, since some bullying probably happens because it's more "reasonable" to be a bully when your circumstances are worse.