Since you can’t contact parents and there are confidentiality issues – call the bully out on it. Point out each time you see him/her bulling another child (I would point it out in front of everyone, so they feel shame for what they are doing). Make sure they understand how disappointing they are for doing such things. Bullying is NOT okay and child bullies are bad people in the making, so I don’t see the need to be gentle with the bully’s feelings.

^{Keep in mind that I have absolutely no authority on this subject since I don’t work with kids and don’t have any of my own, but I’ve seen this tactic work with one friends and their kids bullying other friend’s kids.}