Neither

I understand perfectly and I'm not trying to be difficult, mate. The discussion was over Costco's warranty. You said it was better than any watch company's warranty. I'm asking if you can back that up or if you're just speculating. You've established that if the customer is willing to pay for the repair that OP will indeed repair it without proof of where it was purchased, so that's a start. But I still doubt anyone at Costco is going to send it in for you and I also doubt that Costco would reimburse you if you sent it in and paid for it yourself. My guess is that they'll offer you a replacement or refund.

You were the one that made the declaration that "Costco warranty is as good as it gets, Better than any watch company will give you." So I was hoping that you had some information to support that assertion. I don't doubt they'd make it right by giving you a full refund, but that doesn't help if it was a watch with sentimental value or a discontinued model.