I have no idea about shipping from NZ, But I have done quite some research on buying a bike compared to shipping from Europe.

1 buying bikes versus renting

Renting
a) Cost: renting is good up to 3-4 weeks. I think that is a common conclusion. If the trip is longer. Buying is preferred.
b) Riding: A rented bike can never cross a border. So you option should be to arrange on rental per country
c) Time: renting a bike will save you a lot of time. Compared to buy and sell.

Buying
If you buy a bike in Chile or Argentina. You can't legally ride the bike out of that country. (export restrictions) As long as you are not a resident of that country. (I would expect a list of replays listing options to get around this. But...)

When it comes to Brasil. It was possible to buy and register a bike. And ride it out of the country some years ago. Even it was very complicated. I have not found any update information. I have even written to the Brasil government (In Portuguese).
I have contacted several dealers in Brasil, with very low response.
I do not even get prices. They advertise prices without taxes. And than add taxes for a,b, c and d. And than taxes on top of taxes. It gets expensive.

But Brasil is you only option withing these three countries. To buy and sell a bikes.
It would be good if anyone have access to formal laws. And could publish summary or links.

General for buying and selling, is that it only works if your trip starts and ends in the same country. If we are talking about a bike registered in that country. (E.g for US registered bikes there are ways. But that is mentioned elsewhere)

4 Insurance
If you mean insurance for the bikes. No issue. You buy a third party insurance per country (SOAT) . There is a lot about that here on this hub