Not sure if it’s any help but I have been out here (grew up in UK) for most of the last decade.
The only gun I brought over here was a custom shotgun I couldn’t/wouldn’t bring myself to sell. I shipped it over and am more than happy to PM you the details of the company that help sort it out for me if you want.
As far as bringing guns out I would say the best bet would be to fly out with everything else (binos, scopes, thermals, clothing, sticks etc.) and buy the guns here. Unless you have something expensive and european (think Blaser/Strasser/Mauser) it’ll be just as easy to buy over here and half the hassle.
As someone else mentioned the easiest thing is to borrow a rifle in the short term and apply for the license whenever you can. Clubs will help I’m assuming but I never went down that route. If your coming to the South Island PM me and I’ll help wherever I can.
W