You're not the only one looking for a decent counter service place at the Studios. I've been searching for one, too. The offerings are so bad there that I often work my schedule around not being in the park for a meal, unless it's at a table service location.
Backlot Express is pretty decent, albeit basic. The grilled turkey and vegetable sandwiches are different, at least. The burgers are standard, but there is a toppings bar. Also, the menu at the Studio Catering Co. looks interesting, at least. I went there once and was underwhelmed. Backlot Express is probably a better option, especially with your tyke.