The car is a great way to explore the island. However, in most state camping grounds sleeping in a tent is mandatory. Check if private campings still have availabilities before you book as it is very busy all year round. A big tent is provided which is really easy to set up and is supposed to be waterproof. It doesn't have a cover though so I'm not sure if it leaks (we covered it with our poncho and only had light rain so we were fine). The tent was a life saver for us, we only slept in the car once as it was really small and the police was patrolling the whole night (the mattress doesn't cover the whole width of the car and the car isn't crazy long either, we touched the back door and the seats).
The pick up was really convenient as it is really close to the airport, make sure to book a return flight in the morning as you have to return the car by 10 AM and the airport isn't really close to anything.