Oh I get you! 
I would definitely recommend Lucca as a day trip (speaking as someone who's led a tour out of Cecina Mare!
).
Agree that Florence will be insanely, searingly hot (it's in some kind of shallow bowl in the hills, which seems to make it even hotter than anywhere else), so if you go there try to take it easy and not rush around too much. The Uffizi's air conditioned IIRC.
You should be able to fill one day in Pisa fairly happily by doing the Campo dei Miracoli (where the Tower, Cathedral, Baptistery etc are, all together). I think they have shuttle buses from the centre if it's too hot to walk. It's amazing, but not a great deal of shade there.
If you can cope with another Big Day Trip, Siena is also amazing. (it's on top of a hill, so should be a bit cooler).
If it was me, I'd get a booking.com special of a non central, air conditioned hotel in Rome, near an outlying metro stop and with parking, leave the car there and get the metro into the centre.