I'd suggest go for all 3 of them. Within your CV put a bullet point that you will have your visa by estimated date (regardless of the employer/consultant reading/overlooking it). I don't know about the exact practice while scrutinizing job applications but it doesn't sound like a practical approach to maintain "No" list.....especially while looking at shortage in certain areas.
Start applying now and give them tentative dates of flying into NZ (and also mention that (if selected) you are ready to start role immediately/whenever you are able to)
Once you receive visa stamp, update your cv and in all cover letters add note that you have visa to start work immediately
After landing here in NZ, again update your cv and cover letter with local contact nos. that you have, all consulting agencies make it mandatory that you visit them face to face at least once (that’s to ascertain you are same candidate, gather identification & visa eligibility, complete application for verification/complete application for local criminal records with Ministry of Justice/background, check your confidence level etc)
Hope that helps you to get started, all the best
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