Hi Anne, NSW online Electoral Rolls (available on Ancestry) only begin in 1930..
From Australian War Memorial Service number: 3182
Unit: 55th Battalion (Infantry)
Service: Australian Army
Date of death: 19 October 1917
Cemetery or memorial details: Ypres (Menin Gate) Memorial, Belgium
War Grave Register notes: MULVIHILL, Pte. Lawrence Rupert, 3182. 55th Bn. 19th Oct., 1917. Age 27. Son of John and Ann Mulvihill; husband of Eva H. Mulvihill, of 39, Kensington Rd., Summer Hill, New South Wales. Native of Tumut, New South Wales.
Source: AWM145 Roll of Honour cards, 1914-1918 War, Army
I'm not sure if you have seen the info at Australian War Memorial site but it worth a look. Also has the "letter of notification" sent to Eva when he died.
They should actually have a photo of him as photo's were taken on enlistment,,before leaving Australia.