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 halves, cover, and refrigerate for 2 to 24 hours.
Prepare an outdoor grill with a medium-high fire for bot direct and indirect grilling. Position a drip pan under the grate on indirect side. Place the chicken, skin side down, over direct heat and cook until skin crisps and has definite grill marks, about 4 minutes per side. Move to indirect heat over the drip pan and cook skin side up, covered, until an instant-read thermometer inserted into the thickest part of the thigh registers 165°F, about 35 to 40 minutes. Let the chicken rest about 5 minutes, then cut into pieces and serve.
Pickapeppa -- the celebrated Jamaican bottled sauce -- is a blend of tomatoes, onions, sugar, cane vinegar, mangoes, raisins, tamarind, peppers, and spices. Fans use this "Jamaican ketchup" on all manner of grilled foods. It adds a distinct punch to this version of the island's spicy jerk marinade.