You have him trying to log in via your ID? Or his?
I would have him set up an account under his ID and ask support@chess.com
if they can transfer the premium membership to his ID. They may not be able to because I believe that Apple controls everything.
I subscribed this account via Apple ID. however my son can't login this account from his iPad to play games. (his iPad is under his apple ID. How could I let him access? I don't even play chess on my phone. Thanks!