My believe is that it is not open slather, - you cannot just choose the Regulator you prefer in any country.
I also feel that the aircraft had to operate out of the country for a period to be eligible.
The times I have seen this have been for 'first of type' or other unusual conditions. Even then they eventually went onto the local register.
In this case, in Australia, you will have two regulators to deal with. The aircraft would be on the register of one country and have to comply with applicable rules in force there, - as well as operating iaw CASA regulations.
I have maintained aircraft in the local area using an FAA A&P but only ever on ad hoc and interim repairs.
A maintainer would really need to have an IA to be useful.
For what it is worth it is much easier operating within the NZ CAA rules. It seems to be one of the better regulatory systems about.