Puglia for 7 nights

I have the same question, but we would like to start in Naples/Sorrento to see Pompeii or Herculaneum

We stayed in Martina Franca and found it to be a lovely town and especially pretty at night. From there we visited Alberobello as a day trip using public transportation so I can’t comment on what it is like to stay there, but I would avoid moving around so much which uses up a lot of time, and stay two nights at one or the other. Also, you might want to eliminate the two stops on the way to Matera so that you have enough time to explore Matera.

We have spent several weeks in Puglia in several trips. We made two day trips to Alberobello, once in the winter for a couple hours and again one March because we were traveling around Puglia with our grandchildren. Even out of the busy season, it was plenty crowded and unpleasantly full of tourist shops spilling out into the streets. If photographed from the right places at the right times of day, it looks charming. And it is very interesting despite being so tourist-oriented. But I wouldn't want to spend the night there --- if you want to spend the night in a trullo, the countryside all around is dotted with rental trulli, or you could stay in a trullo at a B&B like Masseria Aprile outside of Locorotondo (an hour's drive from Bari airport).

Everybody travels in their own way, but we try really hard to make day trips from a home base rather than to visit places on the way to other places. This is partly because we hate to leave everything we need for the whole trip in a parked rental car in a town we don't know, and partly to minimize packing and unpacking every day or every other day --- a day trip that takes half an hour to an hour and a half driving time seems preferable because we can explore towns without any worry and stop anywhere we want along the way that looks interesting while our belongings are safe back at the apartment or B&B. You can get to everywhere you want from either Lecce or somewhere in trulli country --- for instance, Locorotondo to Taranto is a 40+ minute drive. And Puglia is a fascinating area to drive around in. So, I'd stick with staying in a trullo somewhere for 2 nights, Matera for 2 nights, and Lecce for 2 nights, with your last night near the airport.

If it helps with your plans, Brindisi airport is very nice and close to Lecce, if you'd like to fly into one airport and out of the other.

You need to also stay in a seaside town such as Polignano a Mare or Monopoli.
Don’t stay overnight in Alberobello! A few hours is enough.
I don’t feel that Matera is worth an overnight and not two overnights. It has a tragic history to learn about.
We spent ten days in Polignano a Mare as our base and loved it there.