US, UA, and DL will allow a stopover on a round trip itinerary. Save the BA for short hops (within Aus/NZ or to position in the US). US has some caveats, but they're sporadically enforced.

I'm thinking about your itinerary as two big flights (US-Australia and Japan-US) with a medium flight (Aus/NZ-Japan) in the middle and some short hops along the way. I think you should try to build a large open jaw of US-Cairns, Tokyo-US to frame the trip, then fill in the rest. While AA allows no stopovers or open jaws, they're worth consideration because their prices are so low. US-Australia would be 37.5K miles and Japan-US would be just 25K miles since your return from Japan would be in AA's offpeak season for Asia 1. That's just 62.5K miles. While Delta may allow a stopover, they would charge 95K miles. With your 32.5K savings, you can pay for the extra one-way that equates to a stopover.

Then use AA/BA miles for Aus-NZ which should run you about 20K miles (side note: I recommend the south island of NZ. Fly straight to ZQN or CHC from Aus). Use United miles for NZ-Japan at 22.5K miles on Air NZ.

This still leaves you with 80K+ AA miles, with your US miles untouched that will probably be combined with AA by the time you need them for your RTW. If you prefer, you could use the Delta miles for this trip and really stockpile the oneworld miles.

Unfortunately, stopovers are becoming very rare in the US-based mileage programs. In your case, the mileage prices are so much better with AA that I would forgo the stopover opportunity anyway.