After the 355 day input, probably the best chance of seeing more seats is either after a Sale has ended, and then about 3 days before departure. But additional seats can come available at any stage. Personally I think it's best to book any cabin you are prepared to fly in, then keep an eye on the cabin you want to fly in.

Don't forget to check LGW-JFK (though it's not BA's best offering in CW, and there is no First), and consider places likes BOS and PHL which sometimes are easier.