Agree with Ash, just drop the rental off before the ferry and re-hire. Some rentals may be pretty strict. Also be careful on going off the paved road as some would not allow it.
If you want to avoid taking the car on the ferry, you can simple rent the car from Auckland to Wellington, then take the ferry as a foot passenger. Once you are in Picton, you can take a new rental car from that location. There are plenty of car rental agencies to choose from on both islands.
I did exactly this two years ago. There are some rental car agencies that allow you to take the car onto the ferry, and others that do not. I had a car with Avis, which does not allow you to take the car onto the ferry. I booked a car (pick-up: Auckland, return: Christchurch) anyway, and they are prepared for that. You return your car at the ferry terminal, hop onto the ferry, and then you get a new car when you arrive at the terminal on the other island. This is then just one rental contract.
The rental car agency will need to know in advance when you are taking the ferry for this service. In my case, I told them when I picked up the vehicle on the northern island. Since you want to do a round trip, you would have to tell them both crossing dates.
Back then, there was no package for car+ferry available – they had to be booked separately. While having to tell the crossing date in advance sounds like a drawback (as it reduces your flexibility), you have to keep in mind that the ferry is often sold out – so will need to secure tickets well in advance anyway, and you will be able to tell the rental car agency your crossing date(s) at that point.
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