When I first saw your question and saw the "with kids" part, I was going to say Cosmic Rays, kids seem to love that Sonny Eclipse animatronic that pops up and sings, and with all the varying food stations, you could perhaps put together a somewhat less greasy meal.
But after hearing the "no greasy" option you'd like, I'd have to agree with the people mentioning Columbia Harbor House. Yes, they have fried chicken strips and fried fish, but I got a tunu sandwich there last week, served on whole grain bread, with white tunu and a tomato on the sandwich, definitely not greasy. I ordered the half-sandwich/cup of soup option and got the vegetarian chili for the soup, lots of veggies in it, seemed less like chili and more minestrone-ish (without noodles). Again, not in the greasy food category!
Hope you have a great time!