Error in a listing I sold - authentication question

Looking for opinions. I sold 2 cards that “go together”, 2 Prime Numbersakes. There was an error in the listing as I marked it as a single card, and hence, has to be authenticated due to the price it sold at. I didn’t realize 2 cards was considered a “lot” on eBay. My mistake and now I know 😆.

I did more research and now know that 2 cards will not be authenticated and will in all likelihood be sent back to me.

Looking for opinions on the best way to proceed. Send to authentication anyway and wait for the cards to come back, after which the buyer will be refunded? Or cancel the order, refund the buyer and relist?

The buyer seems like a good dude and has been cool. I know this is my error and I’m just looking for opinions. Thanks in advance.