As above. Is is possible to split the family group up into two, as this may be easier to find tickets. Be flexible, and consider redeeming points on other oneworld carriers. I'd skip the internet and speak directly to the service desk... it's almost always better.
Depending on where you live, you might find more availability for award seats ex-BNE with the new direct flights to LAX.
Also, a trick that I have used (but it is more expensive use of points) is to book and ticket the separate sectors of a flight (i.e. BNE/LAX on QF 26 I'd book BNE/AKL then AKL/LAX) as sometimes you can score U seats on one or the other - but generally you need to have B class or above to do that.