If it were me, I'd add another Australia stop in Perth or Adelaide, and book PER/ADL to SYD to HKG as a connection rather than try for PER-HKG as a non-stop. Longer flight, access to SYD international lounge. Again, if it was me, I'd also change LAX-JFK-LHR to LAX-DFW-JFK-LAX-LHR. LAX-LHR is a nice long flight, much more enjoyable than JFK-LHR.

I don't have any experience booking with QF, so I can't say if they will book the AA flights as BA codes, but if not, you could book with a travel agent, or call BA and see if they'd do it. I've used American Express travel for a few trips. (With AmEx travel, if you get a good agent you can just feed them flights and they'll book how you want; if you get a poor agent they'll insist on booking everything by searching origin-dest so in that case I just politely end the call and call back.)

Oh, by the way, you have 12 segments now, not 11, because the surface segment SFO-LAX still counts against the 16 segment limit.