Just a few random thoughts on this family.
As there are no marriage records, well any that we can find, for Robert and Elizabeth, I think we can assume they were never married. California Gully may have been just called California [CALI on indexes] and if the registrar was not aware of the name, may have assumed it was California, America - especially many years after the event. I also think that they were both illiterate - hence the inconsistencies with the spelling of their names and dates.
Robert may have jumped ship, and therefore no passenger record.
arrived departed Victoria
occupation = Ship's Cook, Goldminer, Farmer
BISA is usually accurate with shipping records, if they exist. Wonder how long Robert was in South Australia? Was he in a hurry to leave the State???
It would be an interesting exercise to have all the birth certificates to see the inconsistencies with their data. Robert b. c1855 is not registered in either state. I have searched just using Robert and the years 1854, 55 and 56.
Elizabeth was in Victoria in 1853 to register Hannah/Ann Elizabeth who she stated was born in SA. It would be interesting to know
where she actually registered the birth. Did they travel overland to the diggings - Dunolly, Amherst and then finally Sandhurst [Bendigo] or did they come via coastal shipping and travel inland.
Probably not much help but it often helps to pen the random thoughts

Cheers
Cando