The likelihood of criminal prosecution is low, and even if prosecuted, don't expect a stiff penalty. You can't do anything about that anyway. While you can sue civilly, you are only going to get actual damages here and if there was no medical treatment necessary, that is going to be extremely low.
Your best option IS through the school system. I'm assuming this was a public school. Private schools can still allow corporal punishment though if bruises are left, you are wandering back into assault and battery or child abuse territory. You have a misbehaving kid and an assistant who acted inappropriately. You deal with your kid repeatedly and blatantly disregarding a teacher's instructions. If your child is lucky, the school will overlook his misbehavior this time, but this is not a "get out of jail free card". I'm not saying to expect it to be, but having many years of experience in the education industry, wearing a variety of hats, the most reasonable parents who acknowledge their child's faults get much further than those who clutch pearls and claim their angel couldn't possibly have done anything wrong, or go in spitting fire and ranting up a storm. If your wife is a teacher herself, I'm sure she understands this. It can be hard when you are on the other side of the desk and the fault on the other side is so egregious.
I would be surprised if the assistant was retained, but I only have your side of the story, and know nothing about this person's history or how the incident occurred. Your version sounds like this was a deliberate act, in which case I would be shocked if the school didn't terminate, union and all (teaching assistants aren't tenured but may be in a union). In any event, there is no reason not to meet with the superintendent. You won't know what or how the school plans to act until you do. It might be that whatever they propose is satisfactory. There is never a time when it is alright to hit a student. There just isn't. If you were in Texas or of the states that allows corporal punishment, you might have more of an uphill battle, but in NY, I given it is against the law, you probably don't have to go nuclear to get the school to take this seriously.
As for your child speaking with a "shrink", I assume you mean "School Psychologist" or "Guidance Counselor". There is nothing that would prevent your child speaking with them or require you be notified first unless that is a policy of the school (which would be surprising). Did this person say or do something that was concerning?
