HKG-NRT-DEL-NRT-SIN-SYD-AKL-SYD-JFK-LAX-DFW-ANC/SEA-DFW-PHX-LHR-DXB/MCT-LHR-CPT-JNB-HKG

But wouldn't flying to Dubai or Muscat be considered flying out of Europe? I still need to do LHR-CPT afterward.

No, since OW defines the continent "Europe" as Europe/Middle East/North Africa and this includes DXB and MCT (as well as a lot of other destinations which cannot be considered as "European" from a geographic point of view; e.g. Tel Aviv, Cairo).

Thanks for the heads up. :)

So, within Europe, what destination would get me the most miles from LHR? I want something with F service, preferably.

BA flies First between LHR and MCT.

(This is very useful for QFFers as it's just over 3600 miles and into zone 5 so earns 180 SC's E/W in First. - 120 in J)

some futher North America alternatives:

As proposed:
...LAX-DFW-ANC-DFW-PHX-LHR...13,459 miles (if your dates correspond to ANC flights)
...LAX-DFW-SEA-DFW-PHX-LHR...10,693 miles (proposed SEA replacement)

Other possibilities:
...LAX-DFW-MEX-YVR-MEX-LHR...12,608 miles (alternate version, with JL on MEX-YVR-MEX)
...LAX-ORD-MEX-YVR-MEX-LHR...13,868 miles (slightly longer alternate version, transiting ORD instead of DFW)

caveats:

1) JL-flights YVR-MEX only operate twice a week (but will have much better comfort and service than an AA-757 to ANC or SEA)
2) ORD-MEX seems to be a seaonal flight. DFW-MEX operates year round.
3) I didn't check to see if MEX-LHR has F, though I suppose it must.