Using AA miles you can fly Air Pacific, Air Tahiti Nui, Hawaiian or Qantas, so there are a few options. You should begin to book in March 2012 (330 days prior to your dates) and hopefully you will be able to find something.

While, it would cost you more in terms of miles (30k more), you could fly US-Fiji-NZ-US. The 30k extra is for the Fiji-NZ leg. Of course you could tack on Tahiti as well by spending an extra 30k on your return. Since Fiji/Tahiti-NZ flights are relatively short, you could save yourself 10k miles and fly coach. Please note that you can have a free stopover in HNL if you fly Hawaiian since HNL would be considered the first/last North American gateway.

Your options are limited with BA miles, so I'd focus on AA. And certainly trading for Star Alliance miles is an option since most Star carriers allow a stop-over on their awards. I believe Air New Zealand flies LAX-NAN (so Fiji is a potential stop-over).