First thought is I think you are under rating the 7x57. For most NZ hunting it will do everthing you need provided you reload and can make the most of it. Certainly the 7x57 in a good condition action will out perform the 7m08 with the heavier pills.
I see the 6.5 being too close to the 7x57 so would skip that one.
So my first thought was 243 with a fast twist barrel would cover most NZ hunting out to 500 mtr if you reload. Again it's not that different to your 7x57 so I see the 223 as a viable option for shorter range deer and longer range pest control until you get your 7x57 and other here.
Lever actions, Marlin 336 in 30/30 or 35 Rem, Browning BLR in any caliber from 223 to 30/06, used Savage 99.
So if you can find one scratch both itches and get a BLR in 223. Will shoot with most bolt actions.
Z
NB I get wanting to use the 404,
These last couple of years I have been using mostly a 358 win over my 243, The 243 is so much easier to shoot, quality action, trigger etc, a little more weight and better shaped stock. But the 358 with it's crappy trigger and less ergonomic stock, extra kick is just so much fun and challenging to shoot.
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