I had the same issue at LAX. They would let me in because I had the Reserve, but not my wife. We were traveling D1 LAX-DCA, and the agents insisted that LAX-DCA is not a D1 route (only the JFK flights and international are D1 according the the lounge agents). No amount of arguing, including showing D1 on the reservation, could convince them otherwise. Eventually gave up for the same reason you did. If we had a longer layover I would have charged it to my card and asked Delta for a refund later.