Rinse and pat ribs dry. If desired, remove thin membranes from backs of ribs, using a paper towel for a better grip. (This will make ribs more tender.) Rub ribs evenly with jerk seasoning.
Pour rum in a shallow dish or zip-top plastic freezer bag; add ribs, turning to coat. Cover or seal, and chill 1 hour, turning occasionally.
Light 1 side of grill, heating to medium heat (325° to 350°). Place a drip pan beneath unlit side. Remove ribs from marinade, let drain quickly, and place on grate above drip pan. Grill (with lid closed), turning occasionally, 2 1/2 to 3 hours or until ribs are browned and tender and meat has shrunk back from bones.
Brush ribs with 1/2 cup Sticky Rum BBQ Sauce, and grill 15 more minutes. Serve with additional sauce.