Preheat oven to 325°. Place ribs in a large roasting pan. Remove papery outer skin from garlic bulbs, but do not peel or separate the cloves. Cut off top of garlic bulbs, exposing individual cloves; add to roasting pan. Add orange; cover pan with heavy-duty foil and seal tightly. Bake until tender, 2 to 2-1/2 hours, brushing with 1/2 cup barbecue sauce halfway through cooking.
Meanwhile, in a small saucepan, combine vinegar, sugar and water. Bring mixture to a boil over high heat; cook until sugar is dissolved, about 2 minutes. Let cool completely. Place carrots, radish and green onions in a bowl; add brine. Refrigerate until serving.
Prepare grill for medium direct heat. Carefully remove ribs from roasting pan; discard garlic and orange. Place ribs on grill rack; brush with some of the remaining 1/2 cup barbecue sauce. Grill, covered, over medium heat until browned, 15-20 minutes, turning and brushing occasionally with sauce. Cut into serving-sized portions. Serve with pickled vegetables, toppings and remaining sauce.