I'd sort of assumed that a Fiji in Australia would end up with some sort franchise that resembled the West Indian/African colonies. Not necessarily explicitly racist but very much designed to ensure that the vast majority of electors were White British. In which case Fiji would likely remain a bastion of rural / merchant conservatism until that franchise was expanded.
There are different ways to do this too. You could do the NZ way and give the indigenous Fijians elected representatives but on different terms, so with a local roll and open voting. Which would allow the traditional chieftains to retain power through the normal means, whilst giving the country the ability to pretend to be a multiracial democracy.