Australia is an odd one for this sort of thing. The Parliament sat in Melbourne between Federation in 1901 to when Old Parliament House and The Lodge were opened in Canberra in 1927, but there is no official PM residence in Melbourne. The Parliament never sat in Sydney but there is an official PM residence there called Kirribilli House, which has breathtaking views of the harbour, Opera House etc. John Howard lived there during his tenure and had his daughter's wedding reception there, with security at our expense of course not to mention the use of the staff.
So I would have an official PM residence in Melbourne, something spectacular like Como house.