Alphapapa Interesting situation, and why I try and avoid codeshare flights. Your contract is with AirNZ (marketing carrier), who have sub-contracted carriage to SQ (operating carrier). This can bring issues with paid upgrades, access to HVC or paid seating, and baggage issues etc.
If the suggestion above from
codyc1515 doesn’t work, then you might be limited to what SQ will be prepared to do on the day, which is could be more expensive than buying extra luggage in advance. Or just go with what AirNZ wants from you. Vexing not to have clarity…