Pulse the vinegar, rum, brown sugar, scallions, garlic, chile, Pickapeppa sauce, ginger, allspice, and pumpkin pie spice in a food processor to make a slightly chunky sauce. Heat the oil in a medium skillet and cook the sauce over medium heat, stirring, until the oil is absorbed and the sauce thickens slightly, about 3 minutes. Cool.
Rub the jerk paste all over the chicken, cover, and refrigerate for 2 to 24 hours.
Heat a grill pan over medium heat or prepare a grill. Thread the chicken, scallions, and pineapple alternately on skewers (if you use wooden ones, soak them in water for 15 minutes before threading them). Season with salt. (Leave a little space around the meat so the heat gets to all sides.) Grill, turning, until the chicken is cooked through, 10 to 12 minutes.