I have no idea if this is possible or suitable for you and your family, but we once booked empty student accommodation for a city break which brought the price down considerably and you should all get rooms in the same hall. Obviously it’s fairly basic accommodation but we just slept there, and there were showers, double rooms, TV etc in the rooms plus common rooms for you all to get together sometimes. The food wasn’t special but it was plentiful and nutritious and plenty of restaurants in walking distance for a special meal. No one fussing about noise etc but it was very quiet at night. It might not suit your family at all but it may be an option. Otherwise I know Rome is diabolical for travel but if you did stay further afield the cheaper price might fund a minibus for all of you in and out of the city. It can be nice to have a relaxing base to return to after sightseeing all day.