Driving from Sicily to Basilica/Puglia

So the five days include the drive and the Matera stay, right? How much do you like to drive in one day? (Be sure rental contract allows ferry--I keep hearing that sometimes they don't). Hopefully someone with knowledge of Calabria will chime in.
I have not made this trek, but I have been to Matera/Puglia twice and I adore Matera especially. It's only about 45 minutes by car from Bari airport, so if your outbound flight is not early, it would be fine to stay there. I could do all five days in Matera and not be bored--I'd stay at least a night or two.
Bari gets a bad rap, which I think is outdated, but I have even had Italians criticize it. The old town is great for a night. Our second visit, we needed to be close to the airport and stayed in Giovinazzo, just ten minutes from the airport but a pleasant little place. Polignano a Mare is about half an hour from the airport and very popular. I stayed just a bit further down the road in Monopoli.
I think my ideal would be Matera plus one of the Itria Valley towns--they are all close together, so any one will do. I liked Cisternino. You can also visit this area form the coast--Monopoli/Ostuni environs.
sounds like a great trip!

Are we nuts?

No. It's just a motorway drive that hundreds of thousands have done before you. Do no leave anything in the car and avoid the old the SS106 on the Ionic side.

Ask the Rental company if you are allowed to load their cars on those ferries and if you are 100% covered for scratches, small bumps and broken glasses.

I'd take a train/ferry to Reggio, see the 2 Bronzes and pick the car there. Otherwise, You could take a night ferry from Sicily to Salerno, get a car near the port and drive to Matera visiting Paestum and the Cilento coast along the way. Thus You'd skip the all drive through the Calabrian Mountains. Get an idea of driving times on viamichelin.com

I can partially address your questions. We, too, planned to take our rental car from Sicily (Messina) to Villa San Giovanni to see the bronzes in Reggio Calabria. (https://carontetourist.it/en/caronte/schedule-strait-messina-passengers) We were roundtripping, though, so our circumstances were a little different. In the end, we decided to drop that part of our itinerary because of the time and costs associated with a simple day trip. At the time, Auto Europe had no problem with taking the car on the ferry, but do make sure that your rental company/agency is ok with it; some are not.

Several years ago, we spent 2 full days in Matera, which was enough for us. Once you've seen the sassi, you've seen them. Beyond that, it's the usual churches, piazzas, etc. It was a nice town, not touristy at all, but I'm not sure I'd spend
more time just hanging around. OTOH, the new James Bond movie was filmed there, so maybe things are picking up. I thought Bari was dirty, crowded, and lacking in charm, although others don't see it that way. I'm usually not petty-crime squeamish, but I never felt really comfortable there on that score. We used Martina Franca as a home base in that area. It was quiet and well-situated for various day trips. Or, if you're staying in a sasso in Matera, you might want to bookend your unique accommodations with a swing through trulli country nearer to Alberobello, then leave from Bari.

Basilicata is the region in Italy.

My husband and I did the drive twice, albeit in the opposite direction than you propose, and really enjoyed it. We would definitely do it again. Other than Naples and the Amalfi Coast, we thought the traffic was far less than we encountered driving in the north of Italy. On one trip, we flew into Naples and spent a few days on the Amalfi Coast. We then drove to Paestum, staying there for a night and then down to Amantea for another night. We then drove to Tropea for two nights before taking the ferry to Sicily from Villa San Giovanni. We had about two weeks in Sicily before flying home from there. I thought being at Paestum at night was impressive; eating dinner overlooking the floodlight temples. Amantea was really just a stop to break up the ride. It really depends on how far you plan to drive each day. Tropea is seaside resort town with lots of restaurants. The other time we did the drive, we flew into Bari. We didn't stay in Bari so I can't offer any opinion on it. We drove the short distance to Polignano a Mare which was a lovely little town with a small but lovely old section and a stunning sea. It might be somewhere to consider for low-impact enjoyment at the end of your trip. We went there specifically because I wanted to go to the Grotto Palazzese restaurant. (I loved it because the setting was stunning. My husband couldn't get beyond the exorbitant prices, attitude of the staff, mediocre food and pigeon feathers dropping into his food, so he was less than impressed). But we did agree that Polignano a Mare was lovely. We then spent a few hours in Alberobello before going to Matera for two nights. I personally wouldn't short change Matera. I think it deserves at least a full day. We crossed over and stayed awhile on the Amalfi Coast before driving down to Sicily. Again, we stopped twice on the way down, the first night in Maratea and the second in Pizzo Calabro. Both towns were nice and great places just to savor the views along with the local foods and wines, without any big name sights. Again we took the ferry from Villa San Giovanni, spent time in Sicily and flew home from there. Catching the ferry was easy. No advance reservations were needed.

As others have said I don't think there is anything unusual about your plans.

Google Statue of Christ the Redeemer of Maratea, see if it's going to be on your route and worth seeing as a stretch your legs stop.

Google Calitri, a labyrinth of abandoned and slowly being reclaimed houses of a hilltop town you can wander through. "The town is known to locals as the 'Positano d'Irpinia' because its pastel colors and architecture reflect those of the villages on the Amalfi coast."

7 km from Matera is The crypt of original sin - La cripta del peccato originale, worth planning for if you have time in Matera

The total time you have available could easily be spent just in Sicily. Keeping Sicily to 12 days and then trying to see Matera and Puglia in 5 days is almost certainly going to leave you wanting more. Have you gotten Rick's guide to Sicily yet? I suspect you're going to realize you'll need to make painful cuts, especially in Sicily, before you even begin your trip. If you figure you'll have another chance to visit Sicily and Puglia, that's fine, but I'd prefer to cover Sicily in more depth and postpone Matera and Puglia (and perhaps Calabria) till the next trip to Italy. Needing to rush over so much ground will inevitably lead to more time in the car and less time actually sightseeing.

I don't think anyone has mentioned Lecce yet, a lovely Baroque city in Puglia. But I can't imagine you'd have time for it.

I haven't seen it, but I understand Matera is one of the locations in the new Bond film.

Thanks for the great input everyone! We are tweaking plans slightly but as I'm sure you all know pretty soon with travel time (southern is a pain to get to from the west coast!) we are up to a 20+ day trip. Unfortunately, we have to assume this will be our only trip much south of Rome because of the time involved in getting there. We are at 11 full days in Sicily and feel good about seeing what we want, we have visited Paestum before so are taking it easy on Greek ruins but seeing the Valley of the Temples for sure. I have emailed a couple of rental car companies to confirm that the ferry is OK. Just one "one night stand" somewhere between Sicily and Matera etc. should work. The info that the Bari airport is so close is helpful, probably do 2 nights in Matera and then 3 or 4 nights in another small town maybe on the coast since the airport would be close anyway. Thank you so much to all!!!