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

Most dealers in Brazil are not used to sell bikes to foreign so is understandable your lack of response. It is possible for a foreigner to register a motorcycle but first he needs to have a CPF (Natural Persons Register number) to do so. It is possible to do it online (through the Brazilian IRS website) or at the Brazilian Post Office or one of the branches for Banco do Brasil or Caixa Economica Federal (state owned banks). Although it is not strictly necessary (from a legal point of view) many dealerships do not sell vehicles to those who do not have a Brazilian driver's license (CNH) so it is necessary to see if the dealer will accept your foreign driver's license or not, but like almost everything in Brazil with a little conversation and the payment of fees it is possible to arrange a solution.

In Brazil since the motorcycle sales registration system moved to digital format (the transition was accelerated by the change in the plate model - we call it the "Mercosur plate" - in the country and the pandemic) most of the process is done electronically. If you are going to buy a brand new motorcycle (the more expensive option) you can only leave the country if you have the property taxes up to date (it is usually checked at customs) and the motorcycle in your name, if you buy used it is necessary to transfer ownership (which implies the payment of any fines that the motorcycle happens to have, the payment of the transfer fee, possibly the issuance of a new identification plate and some administrative fees). The process of purchasing and transferring ownership is not difficult, but it can take between 7 and 14 days depending on the state in which you buy the motorcycle, as each Brazilian state has slightly different rules regarding deadlines and procedures.

Currently in Brazil buying new motorcycles is very difficult, there are waiting lines for most models in production (depending on the motorcycle model it can take more than 6 months to receive the motorcycle) largely due to logistical problems caused by COVID and for reasons related to the motorcycle production chain, then the used motorcycle option is the most guaranteed way to get a motorcycle quickly, but motorcycles over 300cc tend to be quite expensive compared to the prices of motorcycles between 125 and 160cc.

Generally speaking, the price of motorcycles in the dealers is already the price with sales taxes included in the final price of the motorcycle, but in Brazil the "vehicle ownership tax" (IPVA, which is paid annually) varies between 1% and 5% of the motorcycle value (depending on the state and the vehicle) and you still need to pay an annual license fee to the government of the state where the motorcycle is registered. Until recently it was mandatory to pay a compulsory third-party insurance (Seguro Obrigatório) but 2 years ago it has been charged and the compulsory insurance is no longer required.

I'm from Brazil and if you need any help contact me by email or PM and I can try to help as much as possible, but buying a medium/high displacement motorcycle in Brazil today can be very expensive, so I believe that even with the cost of shipping, bringing your bike will probably be more interesting in the end. Generally, those who come to South America send their motorcycles to Argentina, which has (or at least had until recently) a temporary importation process that is easier and cheaper than the same process in Brazil.