You can't use the AA lounge unless you are on an AA flight. The Admiral lounge is in the terminal next door so you'd have to go outside anyway. But the Bradley terminal is old and doesn't have any decent lounges. I hear there is a nice VIP lounge for celebrities, but I doubt it has showers and internet access.

If Staff told me I couldn't use another lounge I suppose I would ignore them. They do want to herd as many people as possible to the gate in time though. So I understand their motivation. But you can use whatever lounge you have member access to.