I’ve just been looking into this (off to NZ as well). We’re flying with AirNZ, they’ve changed their luggage allowance since we last used them, now you pay half the price of an additional piece of luggage for your first piece of sporting equipment- for us this works out to £17.50 extra each way. Combined with the standard luggage allowance that gives a total of 46kg for bike and luggage- you can split this so that, for example, the bike could weigh up to 30kg, which is the maximum the baggage handlers will accept, and that would leave 16kg for the rest of your belongings. If that makes sense.
Can’t say about other airlines though.
Regarding letting the shock down, on their website AirNZ say that the max nitrogen pressure in your shock that they’ll carry is 200psi- that seems low to me?