I am interested in seeing if a meet up would work too. Do let us know.
In terms of what to wear, I think NZers are dressier than people from North America in general, but relatively casual.
Our weather is notoriously changeable and fickle. You will need a rain jacket as well as a sun hat and sunscreen (the sun is brutal). We are a long maritime nation so in the most part our weather is a bit breezy and humid, but not everywhere. NZ is bigger than people think - but dwarfed by our Australian cousin. Their weather is hotter than ours, but also unpredictable on the east coast.
I would bring sneakers as well as sandals that can be walked in, plus a dressier pair of sandals or shoes for meals out.
Thin trousers/pants and a long sleeve shirt would suit our climate, or a dress with long sleeves. I find jeans a bit hot in summer but some people do wear them.
In terms of the special coral brooch, I don't think it would cause any problems but I think the risk of it being confiscated would mean it is not worth it. Especially when it is associated with your son.
It is hard with pieces like older fur coats, ivory ornaments, coral etc... - what do you do with them when they are older and heritage? I vote to wear them and treasure them, acknowledging it was a different time.