Check to see if your online itinerary on the LH site (which you should be able to access, even if you booked via an agent) shows a fare rule. Most business class fares today between Europe and NZ (SW Pacific) are routing based rather than mileage based. The good news is that most LH routing fares include going via Asia or North America, and even allow via Buenos Aires for a surcharge.

Via Asia you can go via HKG, SIN, TYO, PVG, PEK, and BKK if I remember correctly. Since this is to protect your booking they should allow you to pick the route that best serves your needs.

Go to the Lufthansa site, and try one way flights AKL MXP on your date of travel. Almost any of the routings offered should be covered by your fare. Then go to your agent and have them call LH while you are there.