I accidentally shipped a product to the same customer twice, over a month ago. What now?

I know this is a long shot, and it sounds stupid even to me (but I’m looking for a solution nonetheless).

I can no longer call the buyer, so I messaged him instead. Does Amazon provide a way to call the buyer, after the Amazon phone extension expires? Or another way of getting my item back, or billing the customer for it? It’s our mistake, but we feel that the customer is being dishonest by not telling us about the error.