Many thanks for your help Guv1976. I won't bother with going to another terminal just to drop off my bag. I'm willing to make the journey if it's under about $40.

I'm also looking at switching to the 11PM flight. It wasn't available when I called but I'll try sometime later. That will allow me more time in NY, and I'm already a bit over 4 hours too early arriving into LHR anyway (meeting up with someone).