Combine black beans, kidney beans, and pinto beans in a large bowl and cover with fresh water; soak for at least 15 minutes.
Heat olive oil in a large pot over medium-low heat. Add mushrooms, onion, bell peppers, and banana pepper. Cook until they sweat their juices and are slightly tender, 5 to 7 minutes. Add garlic; cook 2 to 3 minutes more.
Drain beans and add to the pot with the vegetables. Add crushed tomatoes, diced tomatoes, Bloody Mary mix, and tomato paste. Add chili powder, salt, oregano, sugar, black pepper, cumin, and cayenne pepper.
Simmer over medium-low to medium heat, uncovered, until as thick as desired, about 1 1/2 hours.
