David, I already responded to your (identical) post in the Mavic forums, but for the benefit of anyone else who may be interested in flying in NZ, here's a copy of my response:
For flying in NZ, the most important web site to keep in mind is this map of aviation no-fly zones:
https://app.airshare.co.nz/app/apps/airspaceMap/
Outside these no-fly zones, you can't fly over private property (without the permission of the owner), or over people or roads, or exceed 120m (400 ft) in height, or beyond line-of-sight (yeah, right

. As for public land - it depends. Beaches are usually OK (as long as you avoid people as noted above). However, a lot of public land (including a lot of the really scenic places) are managed by the Department of Conservation (DOC), which bans drones (without prior approval). In particular, all NZ National Parks are off-limits to drones, so that definitely rules out (from your list) Mount Cook, Milford Sound (there's also an airport no-fly zone there!), and the Tongariro Crossing.
This map -
DOC Public Conservation Areas - shows the (ridiculously large) land managed by the NZ Department of Conservation. In principle, all shaded areas on this map are off-limits to drones. In practice, as long as you stay out of National Parks and obvious wildlife conservation areas (seals, penguins, etc.), you may be OK. No promises, though

Again, from your list, flying over the water from the shores of the Lakes (Wanaka, Tekapo, Taupo) should be OK, as long as you avoid the aviation no-fly zones from the "airshare.co.nz" map. The very summit of the Roy's Peak trail appears to a small piece of land that's outside the DOC area. (However, expect a lot of wind in this area.) And according to the DOC map, Cathedral Cove is part of a DOC reserve. However, I see
lots of drone videos from there - so as long as you fly mostly over the sea, and avoid people (good luck

, you should be OK.