I was a student at WZ University for a semester to tweak my language skills, they required bank transfer or cash on arrival ONLY. No credit cards, no union pay, no paypal - no other options period.
I agree that Chinese schools should have other options but the reality is, cash is king over here.
I am sure that the OP wants to do what is going to be the least amount of hassle for him and his child. Less hassle, less stress for the child.
I find trying to care for things from overseas to be stressful because things just don't work the way they work back home. I am not talking just about the language barrier alone but the way the Chinese businesses do things.
When a 12 year old is involved, personally I would find a way to withdraw the cash while you are still in Wuhan and give it to the guardian. I am assuming that since you trust your guardian with your son, entrusting him/her with a few hundred dollars in cash would not be an issue.