Never arrived on a LA flight, but i believe you will arrive at Bradley. Immigration, Customs there. It can take from 20 min from leaving the plane to leaving the customs area up to around 1:30, but most of the time less than 1 hour. No fasttrack for J. Just try to be quick and hope that there are no other flights arriving just before yours. In case they bus you get the first bus and stand close to the driver.
You can drop the baggage after you leave the customs are (turn right, not left to street level after customs).
The AA terminal is just next to the Bradley terminal. Turn right afte rleaving the building, it is a quick walk.
Assume you already have your boarding pass. You then need to pass security (premium entrance on the Bradley side of the AA terminal and be at your gate in time.
2:50 seems to be very doable if your inbound is not delayed too much. If everything goes well you will have 1:30 in the lounge.