Seems to me that your best option is to choose AA for your LHR-JFK commutes. First of all, there are beaucoup flights each day giving you plenty of options about when to arrive/depart (early AM to late PM in both directions). And also a potentially bigger availability of lower fare buckets. Secondly, AA offers MRTC which in my opinion is the single biggest reason for choosing AA for coach class travel. Thirdly, with the frequency of trips you have planned, you should sign up for an AAdvantage platinum challenge (see many threads about this in the AA forum). Basically, you get Plt status (ow sapphire) by flying 10K miles in 90 days. sapphire gets you oneworld lounge access on any international itinerary. Also, plt status on AA allows you to snag the coveted exit row seats.

So, AA seems like a no-brainer to me.