Definitely take your little one down to St Elena at the tip of Venice Island. There are playgrounds dotted all around along with picnic tables. If you time your visit in the evening around 5:00 pm or so you'll see tons of local families out socializing. There will also be tons of kids playing around. There will ALWAYS be a friendly game(s) of football (soccer) being kicked around. I think 2 nights is not enough, personally. But I do love going to the out-of-the-way places in Venice.
There's a small take away pizza place with tables set up outside...you can sit there or take the pizza to one of the public picnic tables or just sit on one of the park benches. St Elena evening family social time is really a heartwarming thing to see. You'll probably get a few stares because not that many tourists ever venture down that far. Most of the tourists never leave San Marco because they think that's "Venice".
We love Venice and especially St. Elena.