In a large bowl, combine all of the ingredients and stir well until fully combined. Cover in plastic and refrigerate for at least one hour.
On a grill, over even radiant heat at 250 degrees F, lay the pieces of alligator meat and close the lid. Smoke and barbecue for about 15 minutes, turning the pieces over with tongs, until they are cooked through.
Transfer to a platter and serve with toothpicks and barbecue sauce drizzled over the top.