No airline in the world can combine redemption tickets with revenue tickets.

NZ can issue mixed cabin itinerary but you will need to call them and pay a manual booking fee, $100 per booking.

As for SQ availability, yes, it's a myth. AV can also not see SQ inventory in this part of the world and I think it's in NZ and AV's interest to not release these seats, given that NZ needs to protect their own revenue and AV has ridiculously cheap redemption rate AU/NZ to Asia.