It would be extremely difficult, I'm afraid- the white population in the ninteenth century was incredibly paranoid about being swamped by Asian migrants, which of course is a relic of Australian nationalism that is now safely in the past.
There's not going to be Asian immigration in large numbers without the gold rush, but the gold rush only happened after the move to self-government. About 40, 000 Chinese migrants arrived during the decades of the gold rush and that was enough to cause panic and restrictions.
I can't see it working if the gold rush happened before self-government, since the demographic boom would only cause a rush to autonomy. You might get a slightly larger population of Chinese and Japanese Australians, but it would be much the same picture.
I just can't see any scenario in a British Australia where you have a rise in Asian migration without a corresponding rise in European arrivals. Racial paranoia is absolutely, inextricably at the heart of the nascent Australian culture at this time, and any British government who was seen as making it too easy for Asians to arrive would run into serious trouble with their colonies. There'd be more Eurekas, certainly.
So- there might be more Asian-Australians, but I think they'd have an even worse time of it. It would probably drive even more violent racism, which isn't nice to think about.