I brought a Duchess from Australia and operated it here in California with its Australian registration for some time before putting it on the US register. I'm not sure why you'd prefer to do that rather than get an FAA validation of your Australian License. Validations are very easy and there is also no equivalent of the ASIC here. An FAA A&P mechanic can do most maintenance on an Australian registered aircraft while it is in the USA. I did that on my Duchess. An IA would be the only additional requirement for major repairs or modifications and annual inspections.
While you can't get a validation of your Australian license beyond an FAA private certificate, you can get an instrument rating and anything else you would need. In any case, you would not be able to use a CPL without the right to work here in the USA.
PM me if you'd like to know more. I'm an Australian who moved here about 8 years ago. I operate a flight school and aircraft maintenance business in Northern California. I have a US ATP, CFI, CFII, MEI, A&P certificate and an IA.