When I was nursing and DS was younger and we did dairy free I did this a lot and it was not so bad. I just had to think ahead and have some go to meal items I could ask for specifically. Fast food is harder b/c almost EVERYTHING has dairy in it somewhere.
Most places carry olive oil so you can ask for food to be prepared with that as opposed to butter. Fried eggs, pastas, veggies, plain chicken, etc. can be made with olive oil and some seasonings.
You can buy dairy free breads and make classic kid sandwiches like PB&J and bring along. Most kids won't notice the other being "different" because that is something they would typically eat so won't be a big deal that it's not purchased there.
You can bring premade grilled chicken or pasta salad if you have an ice pack, etc. Salad can be with an oil based dressing rather than dairy based. Canola mayo over here does not have dairy by Hellmans and our health food store has non-dairy organic choices that I have used for tuna etc.
Fruits are always good.
Bring lots of snacks as you said but some can be added to a meal so make it more complete looking and filling if you can only get one main course item that you've thought about ahead of time.
I really did ask for things to be made special and explained there was a food issue for DS and most people were more than happy to oblige.
Best of luck!!!!