Heat 1 1/2 tablespoon oil in a large skillet over medium heat. Add onion, red bell pepper, and jalapeno; cook and stir until softened, 5 to 8 minutes.
Heat remaining 1 1/2 tablespoon oil in another skillet over medium heat. Add venison; cook and stir until lightly browned, 4 to 5 minutes. Stir in black pepper, cayenne pepper, grill seasoning, garlic salt, and Worcestershire sauce; cook until venison is browned, 4 to 5 minutes.
Combine tomato, barbeque sauce, ketchup, tomato paste, and hot sauce in a food processor; pulse until sauce is combined.
Combine onion mixture, venison mixture, and sauce in a saucepan. Simmer until flavors combine, 10 to 15 minutes. Divide among buns; top with Cheddar cheese.