Rome to Northern Italy in Late September

Lucca is better for day trips because you have a lot of Tuscany within 1 hour from there, including Florence. If you rent a car be aware that the historical center of Lucca is restricted to traffic, therefore you need to choose accommodations carefully outside of traffic restricted zones.
A car is useful for visiting countryside and small villages, but if you visit towns and cities, like Florence or Pisa, it's best done without a car. A car is not needed at the Cinque Terre, therefore you could return the car in La Spezia and proceed the rest of your trip by train.

It will be a little less time getting to MXP from Lucca than to FCO from Lucca.

If you find parking outside the walls and can stay inside the walls in Lucca, it's a lovely town for strolling, concerts (Puccini's city), food, and historic sites. Pisa is nearby - if you go, visit the Camposanto (cemetery) and the other buildings in addition to the bell tower. In Lucca, you can walk or bike the walls, have a picnic at one of the garden areas along the walls, climb towers.

I would recommend staying inside the walls in Lucca. I usually stay at San Frediano, a converted convent that backs up to the back wall. Lots to do in Lucca. Walk or bike around the walls, go to top of tower (the ones with trees on top...I forget the name), take a cooking class, great shopping. The hotel is close to via fullingo (the main shopping street).
Karren

I'm not sure why you want a car. But do not think it's necessary for this area. If you were

Traveling into Tuscany, that would be different.

Train from both Pisa> Lucca and Lucca>CT.

For day trips from Lucca, Pistoia is a 40 min. train ride.

From Pistoia you can then travel to Montecatini in 11 min. and back to Lucca in 32 min.

This makes a great day trip and you don't have to worry about parking.

Florence, another day trip, is 1H19 minutes by train from Lucca.

I wouldn't stay in La Spezia itself. Pick one of the 5 cinque terre towns.