Other than Chuck E Cheese's, I'm unaware of any other "child themed" restaurant in the area. Some other options.....
If you could change your hotel venue, Great Wolf Lodge, across the street from the Gaylord would be ideal, but you have to be a hotel guest to enjoy their entertainment and facilities. http://www.greatwolf.com/grapevine/waterpark
A couple of other ideas - date dependent - is Thomas the Train and Nash Farms. This would require you "self-cater" treats and eats.
Again date-dependent, but Grapevine Vintage Train makes round trips to Fort Worth's Stockyards (great for kids and adults) and a shorter "Fun Run". They offer a birthday "package" and will transport a cake, but do not allow any food/beverage to be taken on or off the train. https:/…Default.aspx
You didn't say when you'd be here or if somewhere in Dallas or Fort Worth would be acceptable. Magic Time Machine is in Dallas (Addison). It gets terrible reviews, but it's been going for several decades, so I guess they're doing something right. Medieval Times in Dallas (just north of downtown) might be worth consideration as well.