If you have the time, do an all-NZ round-the-world fare with a stopover in PPT. On their RTW fares they allow the codeshare flights on Air Tahiti between LAX-PPT and PPT-AKL. Then, for the shortest trip back to LAX, add HKG and LHR. They also allow codeshares on Virgin from Tokyo, Beijing and Shanghai on to LHR.

I can't say that I know if NZ RTW fares ex-USA are as cheap as ex-UK, but I paid GBP1210 including taxes for April for an itinerary that went LHR-LAX-PPT-AKL-SYD-NRT-LHR. So anyway, it's more than USD1200 but a good deal for all those destinations.