I can only speak for myself but you have not offended my sensibilities! I think it’s great you are planning this anniversary trip and hope you end up loving Italy as much as I do. ;)
When I said “train to Florence” I didn’t mean fly to Florence. I meant take the train to Florence after you arrive at FCO. There is actually a 1:53 direct from FCO to Florence that you might be able to make. Don’t buy that train ticket in advance though just wait til you get there. There are plenty of other trains as well- but just that one is direct (no changes) from FCO. Otherwise you simply take the LE to Termini and grab a fast train there. There are multiple trains every hour.
Yes- car rental is advised for the Tuscany countryside but not for Florence.
Driving in and out of Florence a few times will be stressful add costly so that is why I recommend just staying there.
Florence is extremely compact- easily walkable. Almost the entire historic center is pedestrianized so you sure can’t drive there!
Have your Bistecca all Fiorentina while there. Stay 2 nights- first night is jet lag, next day see the sights and have that steak dinner.
Then pick up car as you leave Florence.
(There are very few places to rent a car in Tuscany- mainly Chiusi, Siena, Florence- there are a few others but not near where you are going)
Your rental options leaving Florence are much better than picking up a car in Chiusi- more availability/better open hours. If you want to drop in Chiusi your option is Avis/Budget.
Smaller rental agencies are closed on Sunday, close at noon on Saturday and for a few hours mid day during the week.
Fly to Rome- take train to Florence on this day
Florence 2 nights
Pick up car-head to Montepulciano- you are rested and not jet lagged much better than driving after an overnight flight. The drive from Florence to Montepulciano is beautiful- take the Chianti road to Siena then backroads on to Montepulciano.
(If you are worried about driving out of Florence -this is when ZTL, bus lane violations often happen- you can take tram or taxi to the Florence airport and pick up there- your choice)
Montepulciano 6 nights-bike ride, explore the Val D’Orcia- Pienza, Montalcino, San Quirico, thermal baths, relax! etc.
Drop car in Chiusi- (this would be Tues 5/12 check those hours)- and catch the train to Rome
Rome 2 nights- you will arrive in the afternoon, check in- wander do the RS Heart of Rome walk, enjoy the fountains, piazzas, Colosseum al lit up.
Next morning (Wednesday 5/13) do your Vatican tour then spend rest of day more wandering, eat gelato, apertivo etc.
Thursday 5/14 Fly home from Rome.
Just wanted to mention that you won’t be the only people in the Sistine/Vatican museums- plenty of other tour groups offer early entry- and you could save some money here
Check Tour Guy, Walks of Italy, Through Eternity