How should we split 16 nights between the East and West coast of Italy?

Well, my first thought is that 10 days is way, way too short for Sicily. I'd say that even if you had a car. I willingly skipped all of Agrigento, Selinunte and Segesta--which I think few others would do--and found 17-18 days without a car not to be enough, However, it sounds like you'll be making more trips to Italy, so you can trek back down to Sicily then.

Puglia's another area where being dependent on public transportation will slow you down. I haven't been to either Bari or Brindisi, so I don't know how convenient they will be for side trips.

I found Lecce very handy for that purpose, because at the time of my 2015 visit (who knows about 2022?) the Lecce tourist office was arranging one-day van trips to a lot of the popular small towns in the area. The trips weren't really tours, just transportation from town to town. As you may know, it is normally quite difficult to visit more than two small towns in one day if you don't have a car of your own. Those Lecce T.O. full-day trips covered from four to six towns; there was even one that included Matera in Basilicata. The complicating factor was that the trips were not scheduled; they depended on who signed up for which excursion on which day. I went to the T.O. early on and told them I was interested in full-day trips, then I checked late in the afternoon to see whether they had slotted me into something for the next morning. I think I ended up with just one day when there was no excursion being offered. Lecce is a lovely Baroque city, so I had plenty to keep me busy there (including a walking tour). Since there are two of you, arranging a private trip wouldn't necessarily be exorbitantly expensive.

The website of the Lecce Tourist Office is flaky more often than not. At the moment it's all text. For a description of the excursions I took, see the section titled "EXCURSIONS IN SALENTO, ITRIA VALLEY AND MATERA".

http://www.infolecce.it/chi-siamo/703.html?jjj=1639263168799

I must caution you that you are likely to run into seriously not weather in both Sicily and Puglia at that time of year. It's pretty similar to the time of my trip (June 16 - July 8). I enjoyed both areas a lot and have frequently recommended them on this forum, but I would not return at that time of year--and I believe I'm more heat-tolerant than average. If you're stuck with your timing because of commitments at home, prepare yourself mentally for the heat and consider buying a cooling scarf (such as a Cool Tie) or something like that.

The website of the Lecce Tourist Office is flaky more often than not

The website of Lecce Tourist office is: https://www.visitlecce.eu/en

The office is in Piazza Duomo #2, inside the bookstore whose site you linked.

acraven: Our dates are fixed as we are meeting our children and their spouses in Sicily for 6 of our 10 days there. We split the 6 days evenly between Ortigia and Catania. Much of this is determined by limited transit options in and out of Sicily for the three couples. (https://community.ricksteves.com/travel-forum/italy/sicily-where-to-spend-3-nights#1836086)

I am encouraged by your comments on Lecce, I like the idea of a couple of tourist office arranged one-day van trips. We usually don't try for more than one or two "sites" for a day, and like to meander around for the rest of the day.

As for the heat -- our first trip was in mid-July and the second in late June and it was hot even for our customary semi-tropical climate here in New Orleans.

Dario: Thank you for the updated link to the tourist office web-site, it looks awesome.

Thanks, Dario. Every time I look for the Lecce excursion information online I get confused. The place I went looked like a tourist office, or branch of one (there were two locations). The info about the excursions is at the link I provided. Perhaps they are not run by the official tourist office after all, but it's hard to argue with the cost (60 euros for a full day).