For 10 days, with two small children, and all your interests, I would do what we actually did, although we did it in March rather than November. However, we've been in Italy for November (and December) both north and south quite a few times, so I feel like I can give some advice about where to go.
My husband and I took one of our sons, our daughter-in-law, and our two grandchildren (one 3-turning-4 and one 5 months old) to Italy, where we had been many times. We took them to Puglia because we know the region well and we knew it would be warm and uncrowded and have wonderful food and have many things that would interest the 3-year-old --- he still remembers this trip six years later.
We flew into Rome, flew straight from the Rome airport to Brindisi, rented a car at the airport, and drove 30 minutes to Ostuni, where we stayed in a charming airbnb with a rooftop terrace in the old town. This part of town was car-free and just alleys and stairs and views and white-washed buildings, and had three of the best restaurants we've ever been to in Italy about a two-minute walk from our apartment. But we could easily also walk down off the hill into the still-old-but-newer part of town to get gelato and run around in the piazza and find pizza for our grandson and buy groceries, etc., etc..
We did day trips to towns like Polignano a Mare, where there's a pebble beach that I know for a fact is stuffed full of people during the summer, but was delightful in the off season. (There are plenty of beach towns in Puglia, but they will not be even slightly cute in November ----instead you will love to visit towns with a harbor and colorful fishing boats like Trani or Giovinazzo.) We visited the train museum in Lecce and went to Alberobello to see all the trulli. After Ostuni, we stayed in our own rented trullo and did some more day trips, then flew back to Rome for a couple days. You could instead drive to Naples, return your car to the airport (we did this on a different trip to Puglia), visit Naples, and then fly back from there (with a layover in Rome or some other city).
Puglia is so easy to visit and the residents are welcoming and not tired of tourists (at least, not in the off-season!). You could stay in Lecce instead of Ostuni, and just visit Ostuni --- the historical center of Lecce inside the walls is larger and flat and thus easier for a one-year-old to walk around in and for you to have a stroller. Lecce is one of the towns/cities in Italy where I've thought I'd like to live. Anyway, consider Puglia for your family trip!