Bay of Naples - where to stay with a car?

If you rent a car, I'd take advantage to stay on the Amalfi Coast. Positano, Praiano, Amalfi, anywhere really!
Actually the drive is not that difficult at all. There are worse in Italy and even here in California. The issue is parking (and traffic) at peak time (summer, especially weekends). So when you find accommodations, make sure you inquire about parking on the hotel premises or at least if the hotel has arrangements with local garages/lots. Last time I was there with a car was a long time ago, but I'm pretty sure the charge for daily parking is in the 25-30 euro a day. I found having a car there, on the Amalfi coast, useful, because if you want to have dinner to another town, buses don't run much at night. But I also want to say that I've never been there in July or August (May/June only), so I would imagine that July/August would be more problematic for parking and traffic snarls. Also lots of locals spending beach time there often park on the side of the narrow road even if it's not legal to park there (Neapolitans have a peculiar relationship with traffic laws), which makes traffic worse, because then the road becomes narrower and traffic does not flow in either direction. That is particularly true on weekends, even in June, so I can only imagine how it is in July and August.
Positano is a gem, albeit more expensive (being so famous). That is where I like to stay. But any village will do.

Thanks for your input! Sounds like with a car it would be better to stay somewhere on the Amalfi Coast indeed. I will make sure to check parking rates before booking hotels/Airbnb.

We will be visiting last week of October so I am hoping parking lots won't be full and the towns won't be clogged with as many tourists as peak summer.

There is a lovely small hotel in Praiano with parking onsite (your car will have a seaview and so will you). It is La Maurella, on the main road between Positano and Amalfi, but the hotel is one floor lower than the street so it is very quiet, faces the sea.

Sorrento is actually quite good for a car. Most of the hotels on the cliffs above town offer free parking in fact.
We stayed at the Hotel Bel Air and the views are amazing! We enjoyed Positano better as a place to stay but finding parking is much harder there and having a car not convenient either.
Even though you like driving, driving from Sorrento all of the places you mention except for Pompeii may be more aggravation and expense than just taking a bus or ferry. The curvy road is not why I would not recommend a car.

Don't "hope" about parking. Make plans before you arrive. There is no significant street parking in these AC towns, and the paid parking garages (seldom "lots") aren't very big. I don't know about the towns with only one road (like Positano) but Sorrento has significant street closures (I'm not talking about ZTLs here, but actual closure) every night in the center of town.

You may be a little off season, but I can tell you that every night in May, we saw cars lined up to enter parking garages at 6:30 when we left our cliff-front hotel for a walk to downtown dinner. (We didn't want to stay out of town on the hill, because we didn't have a car. We cheerfully paid extra for a cliff-view of Vesuvius and the bay, and loved it.) We stayed in Sorrento for the daily public transportation we needed.

Your driving skills will not be "tested" while you are stopped for busses to back and fill in order to get around the curves on the cliffside road, in front of you. Nor when you are stopped for temporary construction lane closures.