score:4
This website claims to offer a full listing of 2403 LPG stations in Australia, including confirmations from station owners and recent users:
https://www.mylpg.eu/stations/australia/#map
https://www.mylpg.eu/stations/australia/list/
LPG cars are definitely a dying breed, so I'd still be cautious, but even so you should have no problems at all if you take the Pacific Highway to Sydney and continue to Melbourne on the Hume. The most direct route via Newell Highway looks more dicey, in particular there's a 500 km stretch from Goondiwindi to Dubbo with no LPG stations listed.