We did two weeks of luggage supported but self guided bike touring in Northeastern Germany (Mecklrnburg) in 2010. I was happy that we rented. The rental bikes were well suited to the road conditions with robust 32c touring tires. My 23c tires on my road bike would have not survived road conditions including the cobbles and rough roads that we occasionally encountered.
Other friends who bike Europe frequently are a retired 747 pilot who flies nonrevenue. He and his wife rent bikes in Germany to avoid dragging around bikes. Do you want to drag a bike around or find a place to store it when you are on the not biking part of your trip?
I have not biked the areas of Italy mentioned, but a number of bike club friends have. (They all shipped their bikes). Their reports include a lot of hills with evening stops in Tuscany ending at the top of a hill. You may need to train doing big chain ring sprints over Florida overpasses. A light weight bike will feel worthwhile on sustained climbs.
The question only you can answer is how much of the trip is bike-centric vs something else. If the purpose is to ride, you will likely be happier on a high quality bike that is a perfect fit than a heavier rental that is only a close fit. If you do rent, consider bringing your own peddles and saddle.