You can get a permit to shoot wallaby in Tassie.
The NT has a couple of hunting reserves where you can go hunting if you don't have access to private property shooting. You need a permit to hunt pigs there, but not on private property.
I didn't know that about buffalo. I actually saw one when I was kayaking on the Roper river, but I only had my 44 magnum with me and the guy I was with didn't want me to make the shot. I'm pretty sure I can get a hunting permit on an interstate firearms licence. Might have to hit you up for spots or a trip when we are there. Should be a for a week or two in early September