those 1000fps air rifles should be classed the same as a .22 rimfire when it comes to licensing etc IMO
i had a crossman phantom 1000fps and shot a lot of possums and did get a couple of bunnies with it to. You really need to get close to what ever your shooting with a slug gun, ideally within 30m i reckon for it to be effective and be able to place the shot well. Head shots are too difficult with such a light weight pellet, always aim for the chest to hit the heart and lungs to make sure you secure the animal. A .22 rimfire projectile is about 3 times the weight and collective size of a .177 slug and travels a lot faster so it's a far better option for ensuring you kill the animal, it's no good wounding it and it running away to die in the next few hours or days. Slug guns do have their place, they aren't as dangerous at longer distance so there are situations where they are a better option but for effective hunting rimfire is best for small game.
i say get your licence and get a real gun. for the $400 you can get your licence and a cheap rifle. it's about $125 for the licence and not too hard to get. You get vetted then it's a one evening course and test which they aren't to harsh on getting right the first time, it's more about understanding safe practices. You need to have your locking device sorted before you get your licence, i used a eye bolt into the wall and have a bike lock to secure mine which costs around $18 total so that's a cost effective way of doing it. then you are open to a lot firearms activities when you have it, duck shooting, game hunting etc.
I had some time off recently & went for a walk in a forestry block that backs onto a native bush block. scored some dog food and we also cooked some up for us and they were quite tasty.