There's no reason why an NZRFA couldn't be set up to own and operate vessels like Aoteroa and Canterbury, the vessels don't need to be owned by the navy.
Really? Please explain.
As Ng pointed out, it would likely be the NZFA, at least initially, until Royal Assent permitted the Royal prefix, in which case it would most likely be the RNZFA...
One very pertinent question would be, who would own the vessels? In fact, this would likely become an even more important than normal issue in a Kiwi context due to the <insert long string of very colourful language here> Capital Charge, which would need to be accounted for, and on paper inflates the budget. Also the issue of ownership and budgeting would likely become very problematic in the future when the vessels reached the end or their service lives. Who would pay for the replacements, and who would set the capability requirements. I could see the NZDF having a requirement for a dedicated amphibious warfare vessel like an LPD to replace the rather half-arsed sealift capability of Canterbury, but that might not be what an NZFA was either willing or able to provide in terms of capability. By way of comparison, HMNZS Canterbury cost ~NZD$130 mil. while the similarly sized but IMO much more capable
Endurance-class LPD in service with Singapore and Thailand cost ~NZD$212 mil. back in 2008.
Then there is the potential issue of who has authority over said Fleet Auxiliary vessel. If the NZDF were to need cargo, personnel, munitions, fuel, or whatever else, transported to a given area, at present the RNZN could just order vessels to do so. Absent clear lines of authority, a civilian crew could decide to refuse to sail, and/or crew members could just quit if they felt the conditions were unsafe and/or they were not being adequately compensated (pay, time off, etc.). As a side note, the potential for a Fleet Auxiliary vessel to be in harms way is very real, as evidenced by RFA Sir Tristram being badly damaged and RFA Sir Galahad (I) effectively destroyed during the Falklands War in 1982.
Then there would also be issues with NZ law and international maritime treaty compliance for transporting certain types of cargoes, provision for crew/passenger safety in the event of a need to abandon ship, insurance, and vessel tracking at sea. If the vessel is a naval vessel, then the RNZN can issue waivers and/or the civilian laws and international treaties do not apply. A civilian authority and/or operator likely would not have that ability.
And lastly for now, just where & how would mariners be drawn up to provide a civilian crew to Canterbury and the upcoming AOR? The addition of just two (2) vessels would be a nearly 2% addition to the Kiwi merchant fleet. If there are already problems with the idea of using STUFT in emergencies due to a lack of available vessels and crew, I really cannot see how one could expect there to just be crews waiting for civilian vessels to serve aboard.
Again, looking at some of the current Kiwi laws, regulations, and international treaties and how they would apply to a Kiwi version of the RFA makes sense, but to expect or just assert that there are no issues is folly IMO.