What Tinkibelle and Maca44 said. If you have at least 150K AAdvantage miles, you can get 2 award tix in coach, and it'll be much cheaper (in money and miles) than trying to use BA miles or QF points.

I'm assuming that you're leaving from LAX and not Louisiana (but this would still work even from LA). If the thought of spending 14 hours on a crowded 747 in coach makes you faint, but you don't have the miles rquired for 2 biz class awards (250K), you could at least improve your situation by breaking the trip mid-way. You could build an itinerary with up to two stopovers, one at the first international gateway, the other one at the first North Am. gateway. You could, for example, fly from LAX to NAN (Fiji) on the outbound, spend a few days there, then on to SYD (or MEL, or BNE). Then, on the return, you could make a stopover in HNL. Alternatively, if you're a bit short on time, you could fly into HNL from LAX, and that would allow you a "free" stopover (i.e., an overnight, but less than 24 hours) in HNL. Then you'd connect to SYD the next day, and make your stopover in Fiji on the way back. You could substitute Fiji with Tahiti, but that would lengthen the flying time considerably.

QF points (especially if you haven't got many) are not that valuable in comparison to AAdvantage miles. However, if you have a smaller amount of QF points, you could use them to fly to destinations which aren't included in the OW routes (e.g., Vanuatu or PNG). Many of these destinations are close to Oz, so you could get two separate awards to use once you arrive in SYD (or BNE, or MEL). Or you could use your QF points for shorter trips in the US (and then you wouldn't have to pay extra fees and surcharges).

Someone else (perhaps on the BA forum) can give you more advice on how to use your BA miles for this trip (or not).