Heat the oil in a saucepan over medium heat. Add the onion and simmer until soft, 5 to 7 minutes. Add the ketchup. Pour the vinegar into the ketchup bottle and shake to loosen all the ketchup from the sides. Pour the vinegar from the bottle into the saucepan and add the Worcestershire sauce, mustard, brown sugar, lemon juice, hot sauce, and pepper.