I would keep ALL your diabetes paraphernalia in your hand luggage. The insulin cannot go in the hold because it gets too cold. But regardless of that, there is less risk of your stuff going missing if it is with you.
Hopefully, your doctor’s letter mentions that you will be carrying sharps. I have only had to show my letter once and have taken close to 100 flights since my type 1 diagnosis but there is somethings some nervousness about carrying sharp items.
I confess I pay on this a little when I check in - the airline staff asks if I have any sharp items in my hand luggage, I say “yes”, watch their look of panic as I hestitate before mentioning the diabetes and see them visibly relax.
I usually bring my sharps home with me but this would depend on how long you are away. I don’t carry an official sharps container: I just put them in an empty drinks bottle which I decant when I get home. The used sharps can go in the hold on the way back without any problems.
Have a wonderful trip in NZ. I admit I am a tad jealous.