I asked my GP and my partner tried with his GP. Mine has the worst front office dragons so no chance to even get a proper answer. My partner's GP is Chinese and quite used to these requests it seems and does it for $240 but that practice has always been very "creative" in their billing and will put it though as two GP sessions each with our private insurance so we will only be $40 short.
Given you most likely don't have a GP here in Auckland, I would recommend Labtests
COVID-19 Pre-departure testing - Labtests who do it for $141. This is the place where as a Kiwi, you usually get sent for your standard government-funded blood tests etc. so it's definitely a source that will be accepted by the authorities. They also don't guarantee the turnaround times but neither do any of the more expensive testing centres and I have several work colleagues and friends who have used them for pre-departure tests and got the results well in time. They also have dozens of locations scattered around Auckland and in my experience, have always been very professional and friendly- not a single botched blood sample in 3 years and I have VERY bad veins due to Chemotherapy in my 20ies. So they'll definitely be able to do a standard Covid nose-tickle test!