Side note: I love when things like that just pop out and you discover "definitions" no one person you know in that area would ever use, like when Milano inhabitants discovered that GMaps decided that there is a Chinatown in the city :-D
Anyway, bttq: the concept of "Italian lake region" cover a surface of about 11.000 km, so maybe you should narrow things a bit. If, however, you are just searching for a camping on a lake in that area, with kids, your best bet is Garda Lake, for a few reasons:
There are many camping site with the usual stuff plus houses, big full houses.
There are many theme park near (for kids and adults)
There are many nice and interesting locations to visit and the panorama/environment is really wonderful. And I'm not from the area.
If instead you want to stay on Maggiore or Como lakes it will be way more difficult, for a simple reason: those are areas for a different kind of holidays, mainly old and/or rich people. Or (and I'm not going into details 'cause it would be too long to explain), just because most of those are areas where inhabitants are not interested in tourism or in earning money.
So, to sum it up:
- Best bet is Garda Lake.
- Forget about Como Lake.
- If you want Maggiore Lake, search on Piedmont (Piemonte) side, to the north, after Intra and Pallanza
- There is a camping at Ghirla Lake, it's really nice and is full of strangers, few Italians know it. They are really good (I'm not affiliated in any way, by the way), maybe you could have better luck there. Note, however, Ghirla Lake is not a place for tourism: you'll need to take the car and move around
- Don't forget you are traveling with kids! Yeah, I know you mentioned that, but it seems you are forgetting about it when it comes to "number of people in a place". When it comes to kids, if laws allow for it hosting sites (campings, hotel, etc.) become very very flexible: before giving up a place you like, ask if they can arrange a couple beds more for the kids.
Finally, just a personal suggestion: I don't know if your kids fit into it, but camping in a tent is usually waaay more funny. As that area is quite safe, and a camping site being even much more safe, have you thought about renting a bungalow with a bit of a garden and to arrange with the camping owners to have kids staying into tents? (into the garden, I mean)