For brine, in a stainless-steel or enamel stock pot or plastic container large enough to hold the roast combine pineapple juice, salt, brown sugar, rum, and jerk seasoning; stir to dissolve the salt. Carefully add pork. Cover and marinate in the refrigerator for at least 12 hours or up to 14 hours, turning pork occasionally.
Remove meat from brine; discard brine. Rinse roast and pat dry with paper towels. Insert a probe-type or ovenproof meat thermometer into center of meat.
Prepare grill for indirect grilling. Test for medium heat above the drip pan. Place pork on the grill rack over the drip pan. Cover and grill for 1 to 1-1/4 hours or until meat thermometer registers 155 degree F. Add additional coals as needed to maintain temperature.
Remove meat from grill. Cover and let stand for 10 minutes before carving. (The roasts temperature will rise 5 degree F during standing.) Slice roast and serve with Mint -Pineapple Salsa. Makes 8 servings.
Place jalapeno pepper jelly in a medium microwave-safe bowl; microwave on 100 percent (high) power about 30 seconds or until melted. Stir in pineapple, mango or papaya, star fruit (carambola), green onion, and snipped fresh mint. Cover and chill. Makes about 2 cups.