Absolutely do not stay in Mestre, if that is what you mean by "east Venice."
If you simply mean near the train station, that is fine as long as you are off the main pedestrian path heading towards Rialto. This area is known as Cannareggio, and if you are near the northern boundary of the island there is a public pool there if that interests you.
Wherever you stay, getting to the airport would be simplest by water taxi. For five people it is still higher than the vaporetto plus bus, or the Alilaguna, but not outrageously so.
An apartment would be much more cost effective than a hotel room (which will be hard to find for five people). You can search for apartments on booking.com. I tried random dates in August and found many, in prices ranging from $600 to over $1000 for 3 nights. You can filter by neighborhood.
How old are your children? That could make a difference in the suggestions you get.