"...to get a government rate at any hotel, you do have to show your gov't ID or gov't MasterCard...."
I travel for govt business a lot and never had to show the ID once. By the way, it is not true that govt rate is always the cheapest rate. Sometimes it is not.
I judst retired after 36 years with the Fed Gov't and I always had a government issued credit card so that was sufficient as ID....and you're right...even Gov't Airfares are the cheapest but they're refundable whereas some of the cheaper rates are not....