My understanding of the LH fare basis mapping is

C = cabin type (Business)
X = discounted/restricted
OX = advance purchase
I = international
CFF = same cabin for full fare/fare family

LH ticket stock in my experience always include the exact fare class per segment, incl. for partners. You may need to request a full receipt on their website to access full ticketing details if your booking confirmation did not include this, but given you booked a discounted business fare, I would not expect it mapping into NZ-C, most likely J, as others have suggested.