There's probably no secret here - I'm crossing my fingers for you! I've found that my kids are more adaptable to changing schools than I expected. You might have to start somewhere that is less than ideal, all the while keeping your eyes peeled for another opportunity.
You can start a TradeMe account before arriving onshore if you email their support, so it doesn't hurt to contact landlords ahead of time...be ready with glowing referrals from previous landlords about your cats and what great tenants you are. If the area is that desirable though, you may have to rely on luck after you arrive. We had trouble securing accommodation while in Auckland, looking at far-flung areas like Nelson and New Plymouth. Most people expect you to be able to come to walkthrough appointments.
We had good luck with long-term holiday home rental off-season, and some great timing with a family that was going on a 3-month vacation and needed someone short-term. These were certainly cheaper than AirBnb or hotels. You might also look into pet sitting for a few weeks, at least while your cats are still in quarantine.