We went as a family from the UK to New Zealand 20 years ago. When we were booking flights, I remember how surprised I was at the wide choice of routes available. Then it dawned on me - because the UK and New Zealand are virtually opposite each other (globe model), you can in theory pick any point of the compass and any great circle route will get you there just as quickly as any other, so the choice comes down to where you want to change planes. We considered LA, Tokyo, Bangkok, Hong Kong, Singapore, Dubai and in the end plumped for Hong Kong. Various members of the family have been back since and taken different routes and my son is out there now - he went via Manilla a few weeks back.
This all makes perfect sense on a globe, but I'm struggling to see how it would work out on a flat earth.
As far as experiments go, unless you have window seats, can't think of any, but if you do, I believe android phones will allow you to use GPS in airplane mode, so perhaps some kind of GPS tracking app might allow to you at least confirm your ground speed and roughly where you are (see if it agrees with what you see on the in-flight route map and through the window). My understanding is that the fuselage will act like a Faraday cage so you'll have to rely on whatever limited signal gets through the window, so don't expect great accuracy. Altitude data will probably be useless. Time in the air should be similar for the two routes (give or take a bit for wind).
What you really want here is a proper flat earth believer to suggest some achievable test you could do which, depending on the result, could change their mind - good luck with that.
Enjoy New Zealand - great place to visit.