If you look past the tourist restaurants you will see hundreds of little local warungs that serve the most amazing food. Try some of these. If the food is Panang style and is pre-made and on display in glass cabinets, pick a place that is busy so that the food turn over is ok. But if it's a place that makes their food from scratch after you order, you will be fine.
These little local places, in my experience will ask how spicy you want your order to be.
The best advice I can give you is not to judge a book by it's cover in Bali. Some of the crudest, most basic buildings will serve amazing local food.
Especially with the kids, I wouldn't advise eating from the mobile carts you will see aroung the place unless you know a local who can vouch for the food that comes off that particular cart. It took Hubby and I quite a few years to work up to eating off the carts and now only do it from the carts where we know the guy pushing it or we know his usual route and where the cart is loaded at the start of his day.