Going to add one alarm in here. Escalating the attack, both from the first to the 2nd, and from shoe to deep bite, is a very bad sign. It needs to be addressed before it gets even worse. The nice thing with dogs who have bitten several times but been very mild and used minimal force to stop/get what they wanted, they typically continue that pattern. Dogs who start escalating are showing a far more serious pattern.
A total vetting to make sure he is okay, and then a behaviorist. I am not aware of many classes that want or accept dogs that bite. I would try to get individual help before doing a group thing.
While I agree with Congomama about how people react to a muzzle, I am fairly sure your dog biting that child is now known and if you aren't sued, you are one lucky person. This dog should not be in public without a muzzle, so whether you simply stop taking him around anyone until you get this behavior sorted, or you muzzle him, taking him out once you know he bites legally opens you to major problems and ethically I don't think you have a choice. While he's not big enough to kill a child, he certainly hurt her, traumatized her and could set of her fearing/hating dogs. You do not want to knowingly expose another child to that danger. And if you have to hang a sign on him that says "I eat rocks if not muzzled" go ahead, but again – I suspect your dog's rep is pretty well known now.