I think your problem is more related to flight availability than the ability to make changes under the fare rules.
You can only include the QF codeshares on TN (Air Tahiti Nui), and they operate Sunday, Thursday and Friday as QF3825 AKL-PPT only. There is no QF codeshare on TN services between SYD and PPT. So you would need to go AKL-SYD-AKL-PPT-LAX.
QF codeare as QF3827 6 days a week PPT-LAX, so that leg should be ok.