Go downtown. There are an incredible number of bars and restaurants around Busch Stadium. I've been dying to checkout Ballpark Village, but I've always got my kid with me when I'm in STL.
Go to Ballpark Village. I don't even know if there's a BBQ place in there, but it looks incredible from the outside and from the pics I've seen of the inside.
It's directly behind the left field wall. You can see the field from the balconies on the right side of the building.