If you want 24/7/365 independent fighter capability you will need more than a squadron of fighters. More like 3 squadrons, particularly with modern 5th gen aircraft and their availability rates.. Particularly if you are talking about longer range patrols, beyond sight of NZ mainland. You will need ~6 refuelling aircraft. You will need E7's. This requires the establishment of many new capabilities NZ just doesn't have. One squadron would pretty much just beable to offer capabilities as Todj indicated, OpFor, training, etc.
Even Australia has to forward base their aircraft, 4000km north east of Tindal, at Butterworth to be even remotely "near the fight" in regards to China. With Indonesia it was different, Darwin was front line, and F-111's had the range to buzz airbases in Indonesia and did so. If NZ is looking at combat fighters, then logically the place they would be located is Butterworth, not Auckland. It would be in a forward defence posture. Not a fortress NZ posture. If the force has rolled over Japan, Korea, the US, Singapore, Indonesia and Australia, I would propose that NZ would struggle to resist against such an overwhelming force on its own.
EEZ is best patrolled with P8's and other long ranged aircraft, and ships, presence is everything.
Part of the P8 is that it is detectable. It gives presence. Being stealthy doesn't particularly help its mission. It is being used for freedom of navigation flights, because literally, everyone can see it on radar doing its job. Its a big, twin engine aircraft, designed for high flight hours.
While unlikely to engage fighter aircraft, there really isn't anything stopping the US integrating SM-6 and having P8 carry it to deter long range bombers. Normally such aircraft would operate well outside the range of fighter aircraft in a hot environment. However, if you are doing freedom of navigation exercises during peace time, they are good aircraft for that mission. Capturing a 737 is not likely to give the Chinese many secrets. Before a P8 lands in China, all the software will be zero'd and most of the hardware trashed. Wearing out China's air force with their fighters and refuellers having to intercept good old 737's works in Australia's favor. Do not assume because China is firing chaff into the P8, doesn't make the P8 a worthless aircraft. Its about rubbing shoulders.
If Australia wanted to escort its P8 patrols. It can.
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That is the obvious and quickest option for NZ. NZ used to operate 6 P3's. Ideally, operating 5-6 P8's will given NZ a powerful deterrent and capability that could come online within 2-3 years, and before the production line winds down. It would make NZ globally relevant in a conflict and enable NZ to assert itself across its EEZ and friendly region where NZ presence is invited. Which may be hypercritical, because US or AU presence may be seen as a country choosing polarity or aggressive. An NZ P8 can be an NZ P8, and AU P8 can't be an NZ P8. This allows a fleet that can still provide capability while undergoing upgrade or maintenance. Australia is clumsy at Pacific diplomacy at the best of time. A pacific nation leader is unlikely to feel like he can express their wants and needs, concerns when Australia is lock step with the US throwing JASSM at China. NZ is more accessible and has a similar strategic stance to those smaller pacific nations.
4 frigates would be an excellent goal. 5 gives sustainable deployment of multiple frigates. 2 currently gives you nothing sustainable, and as you have found with the upgrade, zero capability for extended periods while upgrading and refitting. NZ can afford and historical has operated 4-5 Frigates. Similar sized nations (Norway, Singapore etc) also have operated 4-5 frigates. The enemy will apply pressure when you are at your weakest, which for NZ means when you have no ships available at all.
Modern frigates could even be armed with LRASM or TLAM, giving NZ a ~1500 km+ stand off range for antishipping. The air defence of a frigate will be significant for NZ. Type 31, F110, Mogami, italian PPA, all offer low crew options where NZ could split its existing crew across four ships. I understand the RNZN has a staffing problem, well smaller crewed ships helps solve that. A ship can go to sea much easier if you just need to put a crew of ~80 together instead of a crew of ~200. There is more chance at promotion, more opportunity, more flexibility.
With so many areas that I can see with opportunities to improve on existing skills and capabilities, fighter aircraft doesn't really get a look in. Cool, you get 6 P8's, 5 Frigates, weapons, modernise Army, triton drones, etc, well yeh, maybe some fighters could add to the combined fight, but even then, unlikely to be based in NZ proper.