To obtain an export permit, you will have to have the consent of the receiving government. You will not get that from the current Australian government.
There are times when handguns can be taken out of the country like legit hunting expeditions and international competition. IPSC world shoots are held all over the globe and the arrangements are made ahead of time. They are held in countries that allow handguns to be brought in.
If you were to try to simply box it up and ship it, you will likely face charges both here and abroad. Unpermitted exportation of firearms from the USA is a Federal Offense. It cannot be much better in Australia.
Posing such a question on an open forum is an invitation for scrutiny into your dealings and in bad taste as it can bring undo scrutiny on the forum.
In life, strive to take the high road....It offers a better field of fire.
"Robo is right" Fuzzball