Anchorage airport is one of the largest air freight hubs in the world. Although there are very few scheduled international passenger flights to ANC, there are dozens of widebody (747, MD11, etc.) freighter flights into ANC every day.
If you check with air freight consolidators in NZ, you might find that you can send your bike to ANC as air freight quite easily. The companies that do the flights - Evergreen, Polar Air Cargo, etc. - don't deal directly with 'small' shippers, this is where the consolidator comes in, they put all the small shipments together.
Prices for air freight drop dramatically if the load is not time-sensitive. So, if you don't mind dropping your bike off for shipment in NZ 10 days ahead of time, you might be able to find an attractive rate. You would not be able to go along as a passenger on the same route, as you have discovered already, you would transit through LAX then change planes and fly up to ANC.