Rinse cavity of turkey with cool running water, and pat dry with paper towel. Place turkey, breast-side down, in a deep bowl or roasting pan. In a medium bowl, combine barbecue sauce, vinegar, red-pepper flakes, and sugar; pour over turkey. Let marinate, refrigerated, at least 8 hours or overnight.
Light a charcoal fire, and let it burn until the coals have become white ash. Alternatively, heat one side of a gas grill set on medium-high.
Brush the entire turkey with the marinade. Place turkey on the grill, breast-side up, near but not directly over the coals, or on the side of the gas grill that is not turned on. Cook turkey with the grill lid closed until the internal temperature in the thickest part of the leg and breast is 180 degrees, about 4 to 4 1/2 hours. Baste with the marinade every 30 to 45 minutes, rotating turkey a quarter turn each time. Add coals heated in a firestarter chimney as needed to maintain an even temperature.
When the turkey is fully cooked, remove from the grill, and let sit for 15 minutes before carving. Serve with tomato-herb gravy if desired, or additional barbecue sauce.