I can talk with decent authority about Victoria and New South Wales. I'll try and keep this somewhat restricted so there is room for more questions, as opposed to typing out a novel right now.
Victorian options
For the most part, your easiest option in terms of logistics for turning up here for two weeks and having a bit of a DIY hunt is going to be on public land in Victoria, chasing sambar deer. There are a few fallow deer and red deer in some parts of Vic public land but it's predominantly sambar. There are also some parts where you can hunt wild dogs, but in most of Alpine National Park they're protected.
For this to be your best option you'd organise flights into Melbourne, hire a car, and go somewhere you've researched. You'd need to get your Vic Game (GMA) license and if you do some reading/googling you'll probably be able to find something about an international license. There are no tags, no bag limits, and no 'seasons' as such, but I believe hunting at times that are total fire bans is off limits. Most sambar hunters start getting into the idea around March/April, plenty of people hunt them hard through our winter (June/July/August), but most experienced sambar hunters consider the best time to hunt sambar around September/October. They aren't like other deer species that have one time they rut and it's common to find stags in hard antler at all times of year, but in general terms, September/October is when there is the most rutting activity.
Gates to the higher country in Victoria are shut over the periods there is snow so if you want to go up there you'd need to backpack in. You won't gain much as the snow pushes the deer down anyway. Most of my sambar hunting has been backpacking in to areas on river flats in the middle of winter. Thick country, fairly cold, lots of deer, etc. Come September/October, some gates may be open but snow will be melting so you may find deer in the tops.
Another great option would be finding someone down here with a boat that regularly hunts/fishes Lake Dartmouth. It's a huge reservoir that's full of trout and surrounded 360 degrees by Alpine National Park. I went there last year and it gave me a vibe that I imagine is similar to a lot of what happens in Alaska. Load up a boat (or multiple boats) with a fairly comfortable camp, rifles/bows/whatever, and head out. Find a good spot on the shore to camp and hunt from there. It's fairly remote but it does get decent pressure, but it would be impossible to shoot all the deer there. It's a pretty spectacular place.
Most of the public land available to hunt in Victoria is amazing country.
I will say this - coming over to Vic and hoping to get into the sambar with a stickbow would be an exceptionally big ask. I'm happy to be corrected but to my knowledge, there could be less than a dozen people in Australia who have been able to get after the sambar fairly well with a stickbow. Typically the country is thick and the deer are very switched on. There has been patchy success in recent years where people have figured out they can call sambar in sometimes, but this isn't a magic button by any means.
I'm fairly hopeless as a sambar hunter but I can go into a lot more detail if you need, in terms of what I would do if I were you and I got down here, best strategies to use, best places to look, etc.
New South Wales options
We have a lot of public land here in NSW. I believe there is an international license available. Do some googling around DPI NSW R license hunting and see what you can find. The system is a little harder to navigate than Vic, where you literally buy your license and just go. In NSW you need to book a state forest and have some GPS or similarly compatible device with you. Not hard to work out at all, but just a bit more effort than Victoria. You'll have a bit more variety in terms of game available, but NSW is a big place and some of the forests are a long way out. You could expect to run into goats in plenty of the state forests in the north-west of the state if you gave yourself two weeks to muck around, no worries. Most of the forests along the Great Dividing Range will have fallow deer, foxes, pigs, etc. Cats are everywhere but they aren't as easy to find as out other critters. Some state forests have good numbers of red deer as well. Some of the southern NSW forests have good numbers of sambar deer.
For a NSW state forest hunt you'd either fly into Sydney or Brisbane and organise which forests to go to. Brisbane is in QLD but it's not that far to drive from there to some of the forests in the northern part of NSW.
Again, if you gave yourself two weeks and you did enough research (talked to enough people) you would have a decent chance of seeing a few animals. Goats and pigs are much easier with a stickbow than fallow deer, especially in Sep/Oct as it'll start warming up and the ground will get noisier.
Some of the country in NSW SF's is amazing, but plenty of it is tough to hunt and kind of boring. some of the country is big pine plantations that are choked with blackberry and it's fairly stale. A lot of NSW SF country isn't the sort of thing a North American may be used to where they find a nice big basin and sit at the top with binos and have a look. Yes, there is some country like that, but it's not the majority by any means. Some of the stuff in western NSW is almost entirely flat and the country has more in common with desert than not.
Your best option, in my opinion.
Find someone, or some people, you can build a connection/friendship with. I've been making noise on here and on Archery Talk a bit over the last couple of years and haven't really invested the time I need to in order to pull it off, but also because I figure I'd just save up some money and come over and do a DIY thing in North America for a month for my 40th birthday (I'm 36 in February).
If you can find an Aussie (or Aussies) who can do things like pick you up at the airport/s, help organise licenses, have a vehicle with a lot of base camp equipment, and know some productive spots, you'll be in for a much better time. This would especially be the case for people who have private access in NSW, or in QLD (no public land options in QLD), or even private access in the NT for things like pigs, buffalo, donkeys, etc.
I know that if for example I were to host someone from Alaska in September, it'd be fairly easy as I always have two weeks off work from late September to early October. I have a lot of private land opportunities in northern NSW where I live, where we could quite easily get into some fallow deer, goats, pigs, foxes, etc. The fallow bucks will be dropping their antlers at this time so trophy potential isn't really there but certainly meat options are. Goats and pigs tend to be an all year thing but they are easier to find in some places than others.
Murray cod (native fish) season closes on 1st of September and opens again on 1st of December so that's not really going to be an option.
Trout season closes in June and opens again at some point in October, so you could maybe tack that onto something you do.
The other part of the equation is, what can you offer someone down here besides friendship? If an Aussie were to go to Alaska for two weeks any time from when their hunting seasons start, what could they hope to get after without having to invest big dollars in a guide or having to rely on drawing a low percentage tag?
FWIW, I regularly hunt with stickbows, compounds, and rifles. My mates and I hunt year round and are always getting into game. We have a great time and the opportunities are essentially endless, but that comes from having decent private access as well.
