Refund denied

Two days ago, my son spent around $540 in 5 - $100 inapp purchases. I submitted a refund request and was denied with no notes. I called apple, and the representative looked over my case. She noted that clearly this does not follow my purchase history, and has every sign that these purchases were made without my permission, but was unable to do anything. She recommended I file a credit card dispute - only issue is I used my Apple card to pay for it.

I submitted an appeal, but what are my options. I'm not letting $540 go to waste on in app purchases that I could care less about and were purchased without my permission. What are my best options?