any LAME who wants IA authorization needs to do a short course with NZ CAA, from memory three days.
Not quite as easy as this but what you are saying is essentially correct.
There are several prerequisites for attendance of the IA course and I recall it was having held a NZ LAME Licence for 5 years. They will be pretty strict on this as the IA is basically doing the Annual Review on behalf of the CAA. They will want a senior LAME known to the system and it is a completely different function to the LAME content of the annual.
For what it is worth I have never used a NZ licence within Oz but I have operated the A&P.
Once again you would need to be a FAA IA in this case to certify the annual on a 'N' reg machine.
It will be the Annual / Annual Review that may cause the issues. As mentioned there are any number of NZ IA operating within Aussie already... the difference may be that they are currently certifying on their CASA licence.
I am sure some are certifying NZ reg machines already of course. Vincent comes to mind but they are operating on a Operators System of Maintenance, not at all the same as the Annual Review. Private operators will be operating under Parts 91 and 43.
The IA needs to attend a refresher course every 5 years also, again just a requirement to keep the certificate current.