If you can possibly through-check the bags, do it. Quite apart from the delays involved in clearing immigration, waiting for baggage delivery and then clearing customs, the AA check-in area at LHR T3 can be a complete zoo.
If you can through-check the bags, then you will stay entirely within T3 without having to clear immigration or Customs. Make sure you follow the signs for flight connections Terminal 3, because otherwise you will get taken off to the Flight Connections Centre where you definitely do not want to be. Intra-T3 connections are very easy.