While I understand your need for a casual place, I think a lot of us who have posted here on LTHForum with some sort of STL ties agree that barbecue should not be considered an option. If you're in the state of Missouri that's more of a KC thing. As far as something not too pricey and casual around the Dome, I can think of a couple of places that would fit the bill. However, when I checked their web sites, I discovered that they were closed on Sundays. I would definitely recommend you walk a bit further and go to Schlafly's Tap Room. The food is good, the beer is good and the atmosphere is very friendly. I think anything you order there will be terrific.
Here's the Schlafly's link. Make sure you click on the Tap Room. The Bottleworks is located in a different part of the city (and also has very good food I might add). Enjoy!
http://www.schlafly.com/brewpubs.shtml