If you're looking to minimize $$$ of out pocket then just write off BA from the start. Their 'fuel surcharges' make it unpleasant, to say the least, to redeem miles on their flights. Much better ways to redeem miles than on BA for Transatlantic flights.

I think Delta runs in to the problem of ridiculous redemption levels for awards so they may be out too.

You're probably then left with either Air France or American for redemptions which get decent value. In either case your best approach is to book as early as possible and be flexible in your dates. And if you want more than 2 seats that may be an issue as well.

Good luck!