Preheat your grill to 225°F.
Score the top layer of fat on the pork belly in 1-inch squares. Don't cut too deep, just barely into the muscle—season liberally with the Sweet Rub on all sides.
Place the seasoned pork belly on the grill and smoke until the internal temperature reaches 165°F (about 6 hours). Spritz with the apple juice every hour while it is cooking.
Once the belly reaches 165°F, remove it from the grill, and wrap it in heavy-duty tinfoil with 1/2 cup of the apple juice. Seal the edges of the foil completely and return to the grill until the internal temperature reaches 200°F.
Carefully remove the belly from the foil and drizzle with the apple juices from the foil. Return the pork belly to the grill and brush with BBQ sauce. Cook on the grill for 10 more minutes.
Remove the finished pork belly from the grill and let it rest for 10-15 more minutes before serving.