Of the US-based FF programs, only United allows stopovers. The problem with United for travel to Bora Bora is that you have to fly via Auckland. Since you want to go to Australia, this may not be the worst - go to Australia first, then to PPT via AKL. But to get back to the US you'd then have to fly back to AKL. Plus Air NZ availability is notoriously stingy for NZ-US routes. You could use another currency for the return PPT-US, but then you won't get a stopover since United won't be a round trip itinerary.

Best is AA. They partner with Air Tahiti Nui, which flies LAX-PPT. AA also partners with Hawaiian, so you can fly via HNL to PPT if Air Tahiti Nui isn't available. Delta is also a good option as they partner with Air Tahiti Nui and Air France (which also flies LAX-PPT). Alaska miles could also be used on Air France. Those are your no-fuel-surcharge options. The best of the fuel surcharge options is probably Korean, but they require round trip US-PPT.

Because so few airlines serve PPT, it's a tough award to get. I wouldn't focus on maximizing stopovers as much as getting award seats to go where you want. Since you have UR, which covers Air France, maybe go for AA so you can book Air Tahiti Nui if it has availability. Whichever you don't use you can use for the Aus-US ticket (Qantas with AA miles or NZ/an Asian carrier with United miles).

ETA: you can't book an award flight to BOB. Best you can do is get to PPT, then you pay for PPT-BOB. I believe it's ~$500 RT.