Any reason you can't switch to AA for XYZ - JFK, that would eliminate all hassle because they would check the bags through to LHR for you.

Otherwise, I certainly wouldn't waste my time going back to the SkyClub. If you can't get your bags checked in early at BA, I would just store your bags in T4 as laid out above, take the AirTran over to T8, go through security with your boarding pass which will get you into both the AA and BA lounges that are now open in T8. Spend 3-4 hours there and then go back out and over to T4 to collect your bags and bring them over to T8 for checking them in.

I would also call or check on the BA forum to see if there really is no one manning the BA/AA desks in the middle of the day for early baggage drop