These are the relevant policies: http://www.airnewzealand.com/oversized-items http://www.airnewzealand.com/excess-baggage-canada http://www.aircanada.com/en/travelin...usic_inst.html
Like most airlines, their policies are somewhat vague, particularly in terms of pricing. Ideally you would try to work with an airline which has a specific pricing guideline for double basses, but in this case that won't be possible.
Air Canada seems to have a pretty clear "no double basses" policy which is tied to their oversize limits. They say you can't have anything over 115 in or 292 cm -- that means linear inches (length + height + width). My flight case (standard Tuff-Lite) is approximately 125 linear inches, but if you have a slimmed-down model you might get away with it.
Air New Zealand seems a little more reasonable. It looks like if you're under 32 kg they'll look past the dimensions of the case. For travel between Canada and NZ, they say an overweight (over 23kg, less than 32kg) charge is 150CAD, but they may choose to charge for both overweight and oversize and ask for 300CAD. Either way, this is still cheaper than air cargo.
You will want to have this all ironed out and notated in your itinerary before you go to the airport. Call both airlines and make sure there won't be any problems. If you can pay in advance, that will make things much easier at the airport. Even if they can't make any promises, at least get them to write something about it into your travel itinerary. Having some kind of note about it will make the conversation at the counter a little easier.
I'm not convinced that Air Canada will look beyond their own baggage policies even if you're connecting to a friendlier airline. You can try that argument on the phone, but don't be surprised if they refuse.
I've found it makes things easier if the bass is your only piece of checked baggage. When I fly, I try to pack light and take only what can fit in a large backpack. Hopefully you weren't planning to take your furniture with you.
You also need to be 100% sure your bass will meet their weight restrictions. If it's 32.01 kg, they will have the right to refuse it at the counter. They won't always weigh the thing, but if they do you need to have a plan to dump some weight. If you must, you can remove the soft case and check it separately.
This should go without saying, but always remember to be friendly, polite, and patient. In the end, the decision of what to do with your baggage is entirely up to whoever you're talking to. You can improve your chances by talking to multiple people (which is why you should call ahead, and keep calling if you don't like their answers). At the airport, if you get angry or try to pressure them, they will move on to the next customer. Arrive extremely early (three hours or so) and be prepared to wait. Just don't ever walk away. Eventually someone will be reasonable with you. If you're willing to pay for it, they'll make something work.