That train is almost full, only a few seats left and none together.
You can see seat maps at the trenitalia website by starting a booking.
https://www.lefrecce.it/Channels.Website.WEB/#
As long as you are not blocking other people’s seats, blocking the aisle or otherwise causing a problem for other passengers the staff won’t be bothered if you have your children on your lap rather than their booked seats.
People get fines for not having reservations on compulsory trains or not having validated tickets or correctly filled out passes.