This is common with almost all rental car claims. Loss of use is typically not paid by the insurance company (unless you are getting rental insurance from the rental car agency) unless there's documentation that the car would have been in use had it not been in the repair shop. In places/times where rental cars are not 100% rented this becomes a slight issue. It's a frustrating argument that may best be resolved by negotiating with the rental car agency directly. In your situation it's likely that Payless will never release the data to Amex and you're going to have to pay it out of pocket.