We've been on many European holidays with a trailer tent and then a folding camper caravan and 3 kids. We've camped in Italy, France, Spain, Austria, Germany, Switzerland.
If you are going to Italy, the weather is superb, you just need shade. Most of the time will be spent in an awning and you will most likely be cooking and eating outdoors. I wouldn't worry about the sleeping capacity of your camping trailer. I would just get a larger pup tent where your daughters can sleep. These tents can get hot but a fan on an extension lead and they will be comfortable. This is what we did with our DS.
We've had a series of trailer tent, folding camper and what might best be described as a folding tent caravan. What are my thoughts after 20 years. If you are going to tour then you ideally want something that goes up and down quickly. Nothing beats a caravan especially one with a motor mover and a fold out awning but these are expensive and you need to car with the towing capacity to pull it. Trailer tents take too much time to setup for touring but are handier for moving around tight campsites. A folding tent caravan takes less time to setup than a trailer tent providing you aren't going to put up the awning but you still have to set up the interior of the camper (we had a Pennine Pathfinder). If you are just going to stay on one site for the duration of your holiday then a setup time of 2-3 hours isn't too bad, it doesn't matter what you take as you will be in safari mode (mostly outdoors under awnings).
Camping in Italy is fantastic, it's getting there that's the problem, it's a long way. We went twice staying at Camping il Falcone in Umbria in between Orvieto and Todi. It's on a hillside in an olive grove. Mostly Dutch and Italians. In the most far flung places where we've been the Dutch are always there. They travel far and fast because they invariably use tents and are able to drive faster than the towing speed. The Dutch are a pleasant bunch, civilised and nicely behaved. In our experience, very few Brits get much further than the Dordogne or the Auvergne as they've already had to cross a country just to get to the channel.
Make sure you pitch is large and flat enough. Some of the trailer tents once fully set up with the large awnings might not give enough room for an additional pup tent. The Pennine Pathfinder with awning and extension awning is enormous. We would get many Dutch visitors from other pitches come and look around our tent. It was the hot and cold running water, the loo, oven, hanging wardrobe, curtains, fitted clock and spice rack with down lighters that had them laughing. The Dutch take a very simple approach when camping.