I created this climate map based on the Koppen Climate Classification map. In this scenario Australia is moved 10 degrees of latitude further south and the Lake Eyre basin is filled (shown in map.) Is this an accurate representation of the resulting climate?
Moving any continent, even just the size of Australia, would have unforeseen consequences to wind and heating patterns, and subtle changes to the jet stream, so it is very difficult to tell without running a computer simulation.
From my understanding, current Australian climate zones are heavily influenced by the Great Dividing Range. Its effects don't appear to be playing much of a role in your model. Your predictions might be good for the Eastern seaboard, but I would expect the rain shadow over Central Australia to still be present and significant.