Car on Amalfi Coast?

It could possibly be beneficial if you wanted to get up into the hills, etc., but parking is so expensive and scarce, and May--though certainly more quiet than June--can still include traffic. The driver does NOT get to enjoy the views, at least I hope not, because they need to be watching the buses, as well as for scooters, cyclists.
From Atrani, it is easy to walk the 10-15 minutes to Amalfi to catch a bus or ferry.
Sorrento has bus, ferry, and the commuter train line, which you can use for getting to Pompeii and Naples.
I think it is really a toss-up -- I'd drop the car in Salerno.

Public transportation on the Amalfi peninsula is limited to local buses and ferries. And the buses cannot always be relied upon. If you want to get to smaller towns then you will definitely need a car. But as you have limited time, I would suggest leaving the car in Salerno before going onto Sorrento. Take the train From Salerno back to Naples and catch the Circumvesuviana (separate train, 1 hour)back to Sorrento or catch the boat to Capri, spend some time there and then catch the boat to Sorrento (30 minutes). Sorrento has boats and ferries to all the major points on Amalfi (Positano, Amalfi, bus to Ravello and the archaeological sites by train)and goes to Naples as well. The train takes you to Pompeii from there in 30 minutes. My suggestion would also be to flip the days around spending less time in Atrani and more in Sorrento. Sorrento is a better base for the Amalfi. If you take the train into Naples, I would suggest staying there one night near the port and spend your time in Naples before Sorrento. Naples is great but full-on and chaotic so not the place to end the vacation.

But whatever you do, don't try to drive in Naples! It's a nightmare.