I assume you already have checked the usual suspects.
Try Fiji Airways, maybe they still have sensible rates for pets as cargo: https://www.fijiairways.com/en-us/ab...o-and-freight/. Business class on their new A350 is pretty nice, and breaking the journey in NAN is not a bad thing with dogs in the hold.

AA cargo offers a 50% military discount so if you know anyone in the military you could save a bunch of money (more info). AA only flies to SYD, though.

Flying private across the Pacific will definitely cost you a whole lot more money than even overpriced tickets on scheduled service. You're looking at about $10k per hour for a jet that has the range for this route divided by 12 or so seats.. I`d say expect to pay $15k at the very least.
Also the AUS government has a bunch of BS rules pertaining to animals on board of private jets, too. I mean, restraining them so they can't move from a mat for 16 hours? (click for CASA animal rules). Both broker and charter operator need to be familiar with animal import rules.