You can ask, and they may do so, but as you are travelling on separate tickets there is no obligation for them to do so, and every reason for them to refuse (they become implicated in the baggage tracing/compensation process if your bag goes missing, as often happens for these "unexpected interline" cases)
If they DO tag your bags to your final NZ destination, be sure to bring the luggage receipt to the attention of the gate staff of your NZ flight well in advance of boarding. They will not be expecting any bag for you as they consider you as a passenger originating in SIN who didn't check luggage in with NZ, and you will therefore greatly increase the chances of your bag being loaded onto your onward flight by alerting them in advance to the existence of this interlined bag.