You say you're the chairperson -- isn't there some kind of board? A group that runs things? Is there a procedure already in place for removing co-op members whose children cannot follow rules? If not -- there should be. Surely there is a written contract for expected behavior of the children as well as required participation of the adults in the "co-operative" aspect of running it??
If there IS such a document or contract, that is your tool for ejecting this family. If there is not -- there should have been, long ago, and one is needed now. But more immediately: You are losing members due to these kids. I totally applaud you for your compassion toward this family. I would give them ONE last chance. Have the co-op board -- not just you alone! The board or whatever GROUP runs things -- ask to meet with the mom one on one (absolutely no kids there, hers or anyone' else's). Then the group tells her -- preferably with the written rules right there in front of her -- that her children have one last chance, but even one more incident of her child or children touching another kid, and the entire family will be told to leave the co-op.
I think it's great and compassionate that you are concerned about these kids and recognize that they are only acting out because of what does sound like a hell of a home life. But the co-op is an organization, which should have rules, and consequences for breaking the rules; also, the co-op could be liable, couldn't it?, if a child known to beat up on others is allowed to stay and stay? At a minimum, the co-op is losing members due to these kids' behaviors.
Tell her -- again, not you alone, but the board or governing group -- that you understand the challenges she faces, but the co-op has a responsibility to protect ALL children who participate.
Then you must, must, must follow through. Another incident, they are told to leave. Not asked, told. I feel bad for this woman, but your concern for her defensiveness and anger should not trump your concern for both the other kids and the reputation of the co-op. The co-op will be talked about by those who leave it, and they will say that it allows bullies to stay when others leave. You will then find it harder and harder to get new parents involved. Consider the bigger picture. Be compassionate but only to a degree here because there are other kids who are being physically hurt. ONE more chance, very clearly conveyed to the mom. I would bet money that this family unfortunately will end up being ejected because the kids aren't going to change in time to save their membership in the co-op -- but if I had a kid there I would be focused on the bigger picture and on keeping everyone safe.