The BA arrivals lounges do not sell passes to allow entry. To enter you have to meet the eligability criteria

I would be v suprised if AA gave out Platinum before completion of the challenge. If you are keen to get Platinum early, then find a nice cheap flight to earn you the 600 point before you go or reroute to something like BOS-JFK-LHR ; this would also mean that you would earn a 4029 bonus ( approx ) on the LHR-EBB flight and of course give you business class check in and lounge access for your trip back

Dave