We do driving holidays to Europe every summer, including Italy. We love the road trip aspect, having our own car, not worrying about how much we take, or bring back. Then build in interesting stops on the way. Last year we managed 2 one week long stays at beach resorts, 2 city breaks and a flying visit to a national park. It would have cost a fortune to have done all those trips separately.
We also live a couple of hours from Calais and once, unplanned, did Biarritz to home in one day. It was exhausting but we did it. I wouldnt want to do it again.
Try a site like viamichellin to estimate costs and actual driving times. I find it more realistic than aa route finder.
Don't under estimate the possibility of heavy traffic at weekends during August. Even if the traffic is flowing the service stations can be packed and that ends up taking more time. We find that theres usually some reason for stopping every 2 to 3 hours, even if its a dash to the toilet.
The other thing to think about is what time can you check in on arrival? You dont want to be outside your accommodation at 10 pm and the propect of having to sleep in the car.
I asume you're going for 2 weeks? How about leaving home straight after work, making it across the channel, drive an hour and stay in a motel. BB hotel are good for family rooms. Most of the cheap chains allow you to self book in so it doesnt matter what tme you arrive. Get up early the next day and drive to somewhere a couple of hours from your destination. We have leaving at the crack of dawn down to a fine art. We make coffee in thermos cups, gather up stuff. Get children up and put them straight in the car. Chuck them a blanket and have pain au chocolate to hand if theyre hungry. We then stop for breakfast a couple of hours in. They love it! And kind of doze for the first couple of hours. We can be in on the road by 6 to half 6 easily and get some traffic free driving done.
The next morning get up later, maybe visit somewhere interesting on the way to your destination, if you're self catering get to large out of town supermarket to stock up. Get to accommodation at the earliest check in time.
Could you possibly take an extra half day or day and start out earlier? Like previous years we will be leaving on thurs after work and using the quieter Friday to travel.