Bit late to this but thought I'd add in case of any interest/help.
We used to do this all the time when our dcs were younger, we always did 3 weeks at the end of May/ beg June. They were wonderful holidays and the drive is fantastic. We still go and drive it, but can usually only manage two weeks in the school hols now so don't take the caravan on shorter trips that far down.
We stayed near the coast so choice of beaches, look for off the beaten track ones, e.g. Cala Violina, about a mile walk through cork forest to get there but worth it. Pisa to the north, Rome bit further to the south doable for a day trip and visit to the Colosseum, drive only if very brave otherwise train!
Just trying to remember names of places to visit so excuse any errors. Nice local towns, e.g., Massa Marittima, thermal springs at Saturnia. If you want some more days out, a longer windy drive to Volterra, Siena within easy reach, and San Gimignano, both worth a visit.
The nice thing about 3 weeks is that you can be very relaxed, have a day out, then a quiet day or two, just wander local towns browsing all the terracotta, food markets etc. Everywhere is child friendly.
From memory we have stayed at Valle Gaia, which is a lovely family oriented campsite, I think they have expanded/improved the pool since we last went, smaller and friendly but very good clean facilities.
We've also stayed at Camping Vallicella (their website seemed to be down when I just tried it). Very friendly, some bigger pitches than usual in Italy, lovely restaurant, very child friendly if they are still the same owners. Near a beautiful village, lovely area.
Neither of these is big though, and more on the low key side with entertainment, which suited us at the time.
The majority of other caravanners are Dutch and some German, UK caravans are usually a rarity, large 6 berths like ours even more so so we would always attract attention when pitching up. Plenty of Brits with Eurocamp, Canvas etc though. People are very friendly, we always found everyone comes and chats and helps when you manoeuvre on the pitch, as they can be tight, or the site team will pop you on with a tractor.
A fair amount of pitches available at that time of year though we found it was still best to book ahead and you can usually choose your pitch.
We found spring storms can blow in quickly and can be quite fierce, so be aware of that.