Who told you that the customer is out of luck?
Closing the card... should have stopped the charges. If it didn't and your bank agreed that the postings were in error, and disputed as such by the customer... you should be refunding every penny that comes out from this activity... Consider if the card was lost or stolen, would you all allow the card reported stolen to be used? No way!
However, I do know that from time to time some issuers like AMEX on the credit side will allow recurring charges to post to a closed card (provided your account is still open, like when you report a card lost or stolen)... so long as those charges weren't being dispupted. So I use my AMEX to pay my ADT security bill on a recurring basis... I called my card in lost or stolen, and they closed the old card number... in doing so they allowed those recurring charges for 90 days before stopping them to allow me time to notify ADT of the account number change... but I agreed to that, they asked if I wanted to have that happen.
In life, there is a lot less that could get better and a lot more that could get worse.
MBA Fin/MBS HR
My views only!