Have a look at:
Civil Aviation Safety Authority - Converting foreign licences
It's a pretty minimal process, although the security paperwork is a pain. Much the same all over the western world, unfortunately. Don't leave it too late to start the application.
If you do a bit of web searching, you'll usually find a school in each capital city that offers some expertise in helping foreign pilots to get airborne. Type for type, flying here is still less expensive than in most of Europe. The difference is that, until relatively recently, there has been no substantive equivalent to your LAPL. These days, many schools do both RA Aus and regular GA training and, for a short stay, an RA Aus option may well be cheaper if you have some experience in applicable (lighter) aircraft types. Have a look at the information on the RA Aus and various aero club sites. For example, the Bunbury Aero Club has some quite good summaries of various indicative aircraft options and prices.
Don't be put off by the process - I've enjoyed flying all around the world and have never regretted the effort to get the local licensing arranged.