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If the creditor refuses to accept payment in the form of a cashiers check or money order, then generally they are discharging the debt represented by that payment.
I dont see them as refusing payment, I see them as declining to send an immediate confirmation notice, for whatever reason.
Mail them the payment in a form that provides written proof of your payment and their acceptance, such as a personal check or a money order that is sent CMRRR and/or provides a record of their deposit.
They apparently want to ensure clearing of any payment prior to formally acknowledging that it comprises discharge of the debt.
Regardless, you can establish your payment and their receipt without a separate confirmation letter.