I didnt think there were NZ departures from LAX that late.
I would ask NZ when check-in opens. Because you are stopping in LA overnight you wont have boarding passes already.
First flight from LA is 7:15 or 7:45pm depending on day of week. So likely no one there much before 4pm. However what I am unsure on is whether they will let you check-in at 4pm for a flight so many hours later.