I have just called NZ to check out my options before consulting with my travel agent.

As some of you have stated correctly, NZ won't change the flight, because it was booked via LH. But they said that if it was ticketed by them, they would change the AKL-HND to AKL-KIX and suggested doing this change with LH.

Plus, they offered to change my feeder flight, which was booked with NZ directly, free of charge to match the new connection.

If everything works out like that, I'd be more than happy.