Based on my recent F experiences across OW, I'd recommend you try to take CX or QF (non-380) for the F experience and if you need to add segments to make your 4 BA flights, do a quick LHR-MAN-LHR or some other set of shorthaul flights to get your status. Eurocheat with BA will also help you get to Gold much faster (and definately within the trip you've proposed).

Lately BA has been a bit of a letdown with their F product and service.

If you're going to do an AONE4, you should be able to sample the F product from all the major carriers (BA, CX, QF, JL, AA) which I would highly recommend as you probably won't get the chance to do so again unless the AONE becomes a regular habit...it can happen as I've now switched to ACIRC22&26 rather than DCIRC's