NZ isnt an option - wrong terminal (T2 vs TBIT) and cannot access without BP for flight departing from that terminal.
Assuming you're starting in LAX then ask at check-in which lounge to use. If SQ lounge isnt available (quite likely) they will direct to another lounge for you. If connecting at TBIT then can also ask at the SQ lounge - they will either let you in or direct to another lounge you can use.