Wow, that's a lot in one trip. You might can use US' lax routing rules to get some of the big legs included on one itinerary. A few examples where using US will beat AA:

You can transit HKG en route to Australia
You can transit Middle East en route from S Africa en route to Europe
You can have a stopover in an international destination

So I would try to build an overarching award itinerary with US miles, then use AA miles for one-way awards for the longer fill-in segments, and use BA miles for the short hops especially if economy suffices (i.e., SYD-MEL, SYD-AKL, HKG-PNH, etc).

Oh, and for searching oneworld airlines, my favorite site is Qantas. BA will show the same (I believe), but Qantas you can get a calendar view of availability by class. Just be sure to avoid BA metal on any long hauls!