As others have said it’s a holiday have fun and enjoy! Having said that though I totally understand that you may want to take a cautious approach.
Meat, seafood, eggs, dairy and veg are normally very good here so enjoy them all. Veg suffers in winter though so be warned if you are here then. We don’t import that much so we are more seasonal with what is available. NZ wines are very good.
Most motels here have some basic kitchen things, including toasters which makes breakfast easy. For a good low carb bread try the paleo bread or almond linseed from
https://www.venerdi.co.nz Carbs listed are total not net so two pieces of the paleo bread only have 8g of carbs. It is not sold everywhere but is fairly available. If you like peanut butter try pics peanut butter. PB and toast is my simple go to breakfast when I am traveling if I don't want to cook something or go out. The bread makes good sandwiches too.
In general I have found that more expensive restaurants are easier to find low carb items at. Often you don’t need to do a lot (or any) changing with what is on the menu. The cheaper places here fill you up with cheaper, carbier, food. As to things you can find pretty easily there are lots of all you can eat Chinese places in larger cities in food courts, it is usually pretty easy to select lower carb items from these. Kebab shops are all over the place too (larger cities) and nearly all do salad sorts of kebabs, if there is rice just say don’t include it. Lots of Thai and Malaysian around coconut curries at both should be pretty good and can be quite filling. Add a side of veg if you are hungry.
Smaller towns (and we have a lot of them) are harder so be warned and come prepared if you really want to go low carb then. Still, not too hard to find a place that does steaks and then swap out fries for veg. Breakfasts are pretty good nearly everywhere and often served late so you can always get one for lunch. I get them to swap out toast/hashbrowns for spinach, mushroom, or tomato.
If you go to Queenstown a burger at Ferg’s is worth it, get the giant one. You can eat very well in all of the bigger cities without too much difficulty so you should be fine in any of them unless you are trying to keep carbs really really low.
Have a great trip and happy to help if you have any specific questions!