I made a claim about a month ago, including the link to the competing website. But by the time they got around to verifying the price, the hotel had sold out and no price was listed. So they denied the claim. I called to complain, and they gave me a $50 credit toward a future AmEx travel purchase without actually agreeing to the claim itself.

The next time I'll be sure to get a screenshot with a date/time stamp on it. But given that it takes up to 48 hours for them to verify the price, they could always claim that they had the lowest price at the time. So get a screenshot of the AmEx site at the same time. I still don't know if it will be enough, but at least you'll have the evidence in hand.