Okay, hold up.
By the time the French made any colonial interests in the area, the British were well-established. France couldn't have ever controlled both islands, probably only the south if they'd not made ridiculous decisions. French whalers were in the area around the same time as the British.
If you could somehow get the French there between 1642 and 1769 (between Abel Tasman's fatal visit and James Cook's more successful one) then maybe, yes, the French could get into it, but with the stories from Tasman's voyage? There's a reason they called Golden Bay Murderer's Bay back then. A group of Maori on a Waka came into the bay and slaughtered most of Tasman's crew while he hid in the rolled-up sail. The stories he brought back stopped Europeans from going there for over a hundred years.
The French involvement started in Akaroa in 1838, long after the British had settled New Zealand, two years before the British formed the Colony of New Zealand with the Treaty of Waitangi in 1840.
Aside from whaling stations (which nearly every country in Europe, and the USA, had built in NZ) only the British had made any permanent settlements simply by confiscating land off of the Maori or going to war (see Maori Land Wars if you ever get the chance).
The French always were far more Eurocentric than the British by a longshot and never had ideal relations with the Maori.
A French ship, Saint Jean Baptiste, belonging to one Jean Francois Marie de Surville passed within 20-30 miles of the Endeavour in 1769. He made anchor at Doubtless Bay and traded with the Maori, but after he discovered one of his boats missing he blamed them and went on a rampage. He even kidnapped a chief who had been hospitable to him (called Ranginui) who died of scurvy later on.
In 1771 Marion du Fresne and his fleet anchored in the Bay of Islands and as usual started out well with the Maori, but decided to fish in an area considered tapu (which is like, temporarily sacred due to someone dying there), which enraged the Maori. Their behaviours with the 'amorous' Maori women also spread veneral disease and even competed with the tribes fishing in other areas of the bay. Of course the Maori reacted by killing and eating Fresne and most of his crew.
So overall the French didn't have a lot to offer the Maori, in fact it was the opposite. The British, while making some mistakes (rarely to the extent of the French and usually by accident), were much more culturally friendly and open, realising that cooperation meant a bigger empire.
French New Zealand could not exist at this time, maybe earlier if you can somehow get the French to go before Cook and be nice to the locals. And not get eaten.