Good luck with the death cert., which seems to be your best bet in Australian records at the moment, although the marriage one is usually more reliable because written by the person himself, so not guessed like deaths details often are.

The one who fits the birthday the best is the William who Frances suggested:
[IGI] WILLIAM CESAR BRIDSON Birth: 02 DEC 1858
Christening: 28 DEC 1858 Santon Father: THOMAS BRIDSON Mother: MARTHA WALKINGTON.

1881 census from FamilySearch:
Thos. BRIDSON Head M Male 69 St Anne, Isle of Man, England Farm Laborer
Martha BRIDSON Wife M Female 61 St Anne, Isle of Man, England Laborers Wife
Annie D. BRIDSON Daur U Female 24 St Anne, Isle of Man, England
Cisily BRIDSON Daur U Female 19 St Anne, Isle of Man, England
James QUINNEY Gr Son Male 7 St Anne, Isle of Man, England Scholar
Source Information:
Dwelling Ballakelly
Census Place St Anne, Isle of Man, Isle of Man
Family History Library Film 1342347
Public Records Office Reference RG11
Piece / Folio 5607 / 9
Page Number 11

- Looks like them, with the Cisily?

He was probably too old for WW1 records (which are online through the National Archives), although many people lowered their ages.

Sue