Maybe Canberra,down to Jindabyne for a night or two, back out to Cooma, down to Merimbula via Nimmitabel, and around the coast to Melbourne. All bitumen.
The snowies are probably the most 'different' sight compared to WA. In late May there is a chance of snow, but by no means guaranteed. You will be over the cold very quickly in one of those campers, so the coast might have appeal by then.
The other major route is down the Hume, so dog on the Tuckerbox at Gundegai, submarine at Holbrook, Albury/Wodonga. You can do Canberra to Melbourne that way as a single long day.
Canberra can chew up a few days easily, so allow for this.