we enjoyed eating at the caffe duomo, it sits right behind the duomo. the food was very good, quick service, the atmosphere was laid back, and it was easy to find. and it seems like they would be open on a tuesday being so close to a tourist site.
Most of the upscale restaurants you'll find in the area do have separate rooms used, you can definitely ask for the space to be reserved for a group.
What type of cuisine are you looking for? it generally goes from traditional but revisited Tuscan cuisine in the nicer places to more international fare.
The restaurant Se.Sto at the top of the Westin Excelsior is very nice, more international fare, have separate private rooms so used for conferences and the like. So does the St. Regis across the square.
There are many restaurants near the congress center across from the SMN train station so it is just a question of type of food and how upscale they need to be.
The first place does have that terrace there with view by the Arno, there are tables after but not to the sides by the terrace so you do have the special seating. I would suggest you write to these places and ask, because I do remember seeing tables set apart from the main hall, maybe not in a private room but with privacy for sure.