Heat olive oil in an 8-quart pot over medium heat; brown pork and beef cubes in the hot oil, working in batches if necessary.
Stir in chicken broth, chicken pieces, potatoes, and carrots. Bring to a boil, reduce heat to low, and simmer stew for 1 hour.
Stir in stewed tomatoes, barbeque sauce, green beans, okra, corn, onion, green bell pepper, celery, apple cider vinegar, Worcestershire sauce, garlic, bay leaves, salt, black pepper, and hot pepper sauce.
Bring stew back to a boil, reduce heat to low, and simmer 1 more hour.