A couple of notes.

Bearing in mind that you're paying by the continent, not the mile, and are allowed six segments in North America and four in each of the other continents, you could use the trip to earn BA Gold status quite easily. That would pay you dividends in Avios earned, access to first-class lounges, and many other perks. Attaining that status requires 4 BA-metal segments and 1500 Tier Points (long-haul in F gets you 210 TP, so you can figure the math.)

I too would use CX first class whenever possible, so if it was me, I'd go LHR-YVR (Vancouver BC) - JFK, using Cathay's YVR-JFK flight. There are far worse places to recover from jetlag than Vancouver.

If you want sunshine and Caribbean style, then if it was me I'd head to Barbados or Aruba from New York. BGI is over 2000 miles, hence 140 TP, and Barbados is gorgeous.

I'd also take CX from JFK to Hong Kong rather than waiting around for JAL to release A-class seats (seldom if ever) or riding AA to HND.

So one possible route would be CPT-LHR-YVR-JFK-BGI-JFK-HKG-TYO-HKG-JNB-CPT. That would get you to BA Gold only if the HKG-TYO (NRT or HND) were available in first class, otherwise you'd still be a little short (if my math is right.)

You could solve the problem by adding a couple of segments in Europe - LHR-MCT-LHR or LHR-DXB-LHR are both easy turnarounds that earn long-haul points. And you could add to the North America part of your trip by adding, for example, a couple of flights to the west coast before you head to Asia, for example ...LHR-YVR-JFK-BGI-JFK-DFW-SFO-HKG...

You're paying a lot for an AONEx - it would be a shame to leave segments sitting on the table, as it were.