I said early on in the thread that I wasn't that concerned about the arrivals lounge part of the trip, because I accepted that I may have misunderstood my access privileges based on the AS site saying I could access arrival and departure lounges, and listed LHR among those. But everyone seems to be focusing on that part anyway.
It was the departure from LHR that I really had the issue with. And even upon leaving JFK, the Admirals Club employee wasn't aware that AS members had access. He had to look it up.
My question was really for people who had more experience with the AS/AA lounge relationship than I do, and wondering if and how they have to prove their access privileges. I just don't want to go through all this every time.