Hi, Geoffrey.
Thank you for posting in the Apple Support Communities.
We understand that you want to transfer your son’s game progress, high scores, and game saves from your Apple ID to the new Apple ID that you set up for him. There isn’t really a way to transfer content between two different Apple ID accounts, so he won’t be able to move his progress to his account, but we may have a solution that works for both of you.
You should be able to go to Settings > Game Center on his device and scroll to the bottom and Sign Out. This will sign him out of his new Apple ID, and disable Game Center. When you toggle it back on, you’ll see the option to sign in again with a different Apple ID. Sign in here with your Apple ID (the one where his data is saved). This will allow him to continue playing using your Apple ID, but allow him his own Apple ID for all other services.
The only catch is going to be that the longer he plays with your account, the more progress he’ll accrue, so if he ever does want to switch to his own Apple ID he’ll have more to lose.
Learn more about using your Apple ID with Apple Arcade and Game Center here:
Access your Apple Arcade gameplay data on all of your devices - Apple Support
Hopefully this helps!