Summary: you can study and work to get national certificate in carpentry level 4 with the help of a polytechnic in New Zealand, but its expensive. Other options may be available, i'm researching them and cost cutting options now.
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I have been emailing with tairawhiti polytechnic in Gisborne.
International students can get a visa for the national certificate in carpentry level 4 course. It is 1 year campus based and then you are placed with a builder as an apprentice. cost is 11,820 NZ for first year, and then ' price varies depending on how many units you take.'
Many people, however, acquire the national certificate in carpentry level 4 certificate without ever enrolling in a polytechnic.
here is a link to BCITO training faq
Frequently Asked Questions
and here to BCITO employers faq
FAQ
in above link, read " do apprentices have to go to a polytechnic and be released from work"- it mentions the 'box of books' alternative and another question answered tells about 'what is a modern apprenticeship?'- it is a special support program for 16-21 year olds.
I have initiated contact with BCITO to learn about visa options other than student visa for getting national certificate in carpentry level 4, and i intend to soon be in contact again with Tairawhiti to find out if i could skip the campus based year and still qualify for a student visa.
I intend to post my findings here again.