Anytime you're thinking of making two day trips to the same place that's not, say, a suburb of the city where you're staying, I think you need to be very, very certain it's the right move. The two round trips require a lot of travel time. Trains between Florence and Rome take a minimum of 1 hr. 34 min. each way.
How close will your hotel be to Santa Maria Novella Station in Florence? To the scheduled time on the train, you have to add the time traveling between your Florence hotel and S. M. Novella and between Roma Termini and the sights you want to see in Rome. I'd anticipate 30 minutes extra in each direction, and it could easily be much more (such as to the Vatican).
You also cannot expect to walk into a train station and right onto your train; for one thing, they close and lock the doors before departure time. I always want to get to the station at least 10 minutes early, and that's risky in the case of trips from Florence to Rome and vice versa. The fast trains between Florence and Rome have reserved seats, so the tickets you buy are specific to one train. If you miss that train, you'll have to buy replacement tickets at a probably very high walk-up price. I see that today's price for tomorrow's Florence-Rome Freccia trains is 55 euros one-way, per person. I don't know at what age discounts for children end, but even the two adult tickets would be painful. So I'm betting you'd want a bit more than my 10-minute buffer at the train station.
That 94-minute travel time + 30 minutes traveling to and from train stations + 10 minutes extra at the outbound station adds up to 2-1/4 hours each way, or 4-1/2 hours per day of sightseeing time lost to traveling back and forth.
I'm actually not much interested in ancient Roman sights (I'm more into art and modern history), so I'm not the best one to say whether four days a reasonable allocation for your family. I'd suggest looking at Rick's list of top sightseeing tips for Rome to see how many of them you really want to see. He has such lists for a lot of major destinations right on this website.
https://www.ricksteves.com/europe/italy/rome << Click on "At a Glance".
I think Rick is really good at identifying the key sights from a cultural, historical and visual point of view, but individual travelers need to consider their own interests. (I've never been inside the Colosseum or the Forum.)
Rome is large and the sights are somewhat scattered. You can burn a lot of time just getting from one to another, because Rome's Metro system doesn't cover the city particularly well or densely. To get an idea of how scattered your possible sightseeing targets are, open a map of Rome on Google Maps, then enter the names of the things you want to see, one at a time, and click "Save" after each one. That will give you a map showing how much moving around you'll need to do. It will also suggest areas of the city that might be good places to stay.
