I like the Flagship Lounge at JFK and suspect it will be better than the BA lounge at T7 so it doesn't seem like a difficult decision to me.

That being said, you'll have to do the T3/T5 trek at LHR if you fly AA. Not really that big of a deal but you do have to clear security again and have a 15-20 minute bus ride. Make sure you leave adequate time between your connections.