I would be willing to bet that if you asked your local theater if you could bring in those items, they would let you. You cannot buy them there, so it isn't like you are taking a sale away from their concession stand. At the three theaters that we've used regularly in the last year (one in VA, one in TX and one in WA), the manager has been happy to allow us to bring in carrots, fruit slices, dry cereal/trail mix, string cheese, etc. I simply call and explain that I have young children that are not allowed to eat lots of sweets. The managers are always impressed that I would bother to call instead of trying to sneak things in, and I just tell them I'd love to be able to purchase those things in their concession stand, but until that is an option I appreciate their help.
I set a good example for my kids, they learn to be good citizens, I get to feed them healthy snacks and we get to pass along our preferences to the theater.
A long time ago I talked asked the manager if we could bring in food and explained why. He said he couldn't let food in because he would have to let everyone bring food in. I suggested that the theatre allow outside food and charge a fee for it. I would be glad to pay a fee or pay high prices for good food.