Jetconnect isn’t based overseas as far as CASA are concerned. Jetconnect is QF...using the Qantas AOC. Same Chief Pilot, same training dept, sims etc etc. Jetconnect is mainline as far as the operation goes. The pilots all hold CASA licences. Exactly the same as the Jetstar NZ operation. From a regulatory point of view CASA don’t care whether it’s NZD or AUD that are paid to the pilots.
Can you use Australian Traffic Rights to operate in that manner?
A field day will be had in Parliament over Australian Traffic Rights being used by an operation and employees that don't pay tax here and are not controlled by Australian Law. You might as well just let Air New Zealand apply for Australian Traffic Rights as well while you're at it.