Make the seasoning mix: Combine the paprika, sugar and onion powder in a small bowl. Store in an airtight container for up to 6 months.
Make the barbecue spaghetti: Combine the ketchup, both sugars, 1 1/2 teaspoons pepper, the onion powder, mustard powder, lemon juice, Worcestershire sauce, cider vinegar, 1/3 cup seasoning mix and 1 cup water in a large pot or Dutch oven. Bring to a boil over high heat, stirring frequently to prevent the sauce from sticking to the pot. Reduce the heat to low and simmer, uncovered, 1 hour.
Heat the olive oil in a large skillet over medium-high heat. Add the bell pepper and onion and cook, stirring, until the vegetables soften, about 5 minutes. Add the sauteed vegetables and pulled pork to the sauce and continue to simmer, 1 more hour.
Bring a large pot of salted water to a boil. Cook the spaghetti as the label directs. Drain, then add to the pot with the sauce and toss.