There are four categories of bird found in Australia (mainland and Tas):
1. Vagrants; these are birds from outside the region and are found in ones and twos or small numbers and not regularly, they do not breed.
2. Winter visitors from northern hemisphere, mainly waders; these come here in our summer during the northern winter and do not breed. Some individuals may overwinter.
3. Summer breeding visitors; birds which arrive in Australia from the north to breed in summer. As far as I know there are only two species that arrive to breed and then leave Australia entirely in the winter (Papuan Pitta and Black-winged Monarch). Many species arrive in the south to breed, then fly north in the winter after breeding, but some individuals in these species remain in northern Australia over the winter (even though many individuals of the same spp travel to PNG). So these will also be resident.
4. Residents; breed in Australia, stay in Australia the whole year (with or without local movements).
So, to get the number of residents, get the Birdlife Australia list for the mainland and Tasmania, knock off all vagrants, winter visitors, and the Papuan Pitta and Black-winged Monarch, and all the rest are resident.