When we had dinner at the Napa Rose, our DD was the only child in the restaurant. But she was already 11 at the time, very used to eating at formal restaurants in NYC and did not require a kid's menu.
I did find this children's menu online, but it's not chicken nuggets and burgers.
Napa Rose
Children's Menu
Grand Californian Hotel
Disneyland Resort
Menu Date: October 2008
Children's Menu For Age 11 And Under
Grilled Petite Filet of Prime Beef Tenderloin - with Lemon Whipped Potatoes and Petit Green Beans. $13.00
Penne Pasta - with your choice of: Creamy Quattro Formaggi Cheese Sauce or Fire Roasted Roma Tomato Marinara. $8.00
Oak Roasted Cheese Quesadilla with Sonoma Jack Cheese - with choice of Fresh Fruit or French Fries. $8.00
-- with addition of Free Range Chicken Breast $12.00
Cheese Pizza - with our Fire Roasted Roma Tomato Sauce. $8.00
Simply Buttered Noodles - $8.00
Roasted Free Range Chicken Breast - With Lemon Whipped Potatoes and Petit Green Beans. $12.00
Sauteed Pacific Salmon - with Steamed Basmati Rice and Petit Green Beans. $13.00
Desserts
Balboa "Beach Sand" - A Blue Tahitian Vanilla Bean Creme Brulee "Ocean" with Graham Cracker "Sand", Some Fun Fish and a Beach Umbrella! $6.50
Warm Freshly Baked Scharffen Berger Chocolate Brownie Sundae - with Chocolate Sauce, Caramel, Chantilly Cream and a Cherry! $7.00
Trilogy of Napa Rose Ice Creams - your choice of Chocolate Chunk, Vanilla Bean or Strawberry...three scoops. $7.00
Edited to add: If you do decide to go there, I would suggest you plan on an early dinner time. There's more likely to be other families then as well as the parents that have gotten babysitters because they want a night out from the kids will tend to come later.