I second what
@rifletuner stated above.
Do you already have an specific hunt and outfitter? The outfitter should assist with all paperwork, be sure to ask about the export paperwork.
I did a hunt last October out of Darwin; I flew Qantas from Dallas to Sydney because it's a non-stop with an A380. I would stick with a major carrier, especially with firearms.
There are several travel agencies who handle traveling with firearms and offer assistance with paperwork. I use PWP - 210-686-0766, but I know a lot of guys who use Gracy Travel.
I would recommend getting to your final destination with the least amount of stops; sometimes overnights with firearms can be problematic, other times not.
We did a day of sightseeing (eating, drinking, smoking cigars) in Sydney because the connecting flight to Darwin left too close to our arrival. The hotel across from the SYD airport was both convenient and accommodating of our firearms.