In the Dolomites mountains of northern Italy you will find rifugi (singular, rifugio) where you can spend a night or several. They are a bit remote, i.e., you have to hike into them, for the most part. That might be tough with a baby and all the stuff a baby needs. This is a somewhat typical rifugio at Rasciesa above Ortisei. It is an easy hike of about 30 minutes from the funicular, but still would be tough with all your baby equipment. And no lake, So just consider this an example of the type of lodging.
In another class is the mountain hotel such as Hotel Saltria in the Alpe di Siusi. You can drive in or take a taxi from Compatsch or even Bolzano. It is a full services hotel also providing meals. There is a pool (commonly found in mountain hotels) and a pond (but it is not for swimming).
Perhaps this serves as a small education on some of the mountain lodgings types. It is far from comprehensive. We stay in the little towns, not high up.
An interesting source of lodging is Red Rooster which offers farm holidays in the mountains. Many have a focus on children and may have your desired lake. A car is required for most.
As to vegetarian restaurants, you will find vegetarian meals at almost every restaurant, but not an exclusively vegetarian restaurant. With Red Rooster, you might find a unit with a kitchen so you could cook.
