Is the day you are returning the car the day you switch to your second resort? I'm not sure what you meant by "kind of doing a split stay". If your car rental return day is the same day your second resort reservation begins, then yes, it will definitely work. Your second resort reservation is treated as a completely separate reservation. Pick out a flight which is arriving around the time you will need DME to AKL. Call and give them that flight number and arrival time to book DME.
I'm not 100% if or how it would work if the day you need DME is in the middle of a resort reservation. It may work but it might not.