A possible solution to the QF earning problem, if that's what the OP would still want to do.

This itinerary is close to the AA price, but has the long flights on QF codes. Main downsides: one extra connection on the way out, and it includes a connection at LAX:-



For comparison, the original itinerary as posted by the OP:-



And the much higher cost if booked on QF codes:-



Caveat: These are all generated by ITA, and I haven't checked them against actual QF (or other) website live pricing.