Preheat oven to 425*. In a small bowl combine allspice, pepper, and salt; rub on all sides of tenderloins.
In a 12-inch skillet brown tenderloins in hot oil over medium heat, turning to brown all sides.
Transfer tenderloins to a shallow roasting pan. Roast for 25 mins or until internal temp reads 160*F and juices run clear.
Slice pork and arrange on a platter. Spoon BBQ sauce over the top. Serve remaining on the side.
In a saucepan combine raspberry applesauce, BBQ sauce, and cranberry sauce. Cook and stir over low heat until heated through. Stir in lemon juice. Serve over pork.
Prepare pork and BBQ sauce. Allow to cool for 30 minutes Cover and refrigerate pork up to 3 days or sauce up to 1 week. Warm pork in a 350* oven for 20-25 mins or until heated through. Heat BBQ sauce over medium-low heat until heated through. Serve as above.