"How do I get the AR report and the customer record to match?"

If you have an overpayment from a Payment, you link the Refund by using opening their original payment and applying it to the Check, which is listed as if that is an open invoice.

If you have an overpayment from a Credit Memo but you created the check manually, and the check lists AR, the use the Receive Payment screen. Do not enter an amount; there is no New Money to enter. Here you see their Refund Check listed, and the amount that is an Available Credit; use the Icon on the ribbon at the top to Apply Credits.