When Brave Browser (iOS) > Settings > Media > Enable Background Audio switch is ON, expected behavior would include the continuation of active media streams as legitimate background tasks.
Re “Sketchy streaming site” . . . I would be using the Brave Browser (iOS) Private Browsing Only mode, and also:
How to do a Complete Exit / Quit of iOS apps; and choices of, Power Off, Soft Restart, or “hard” / “forced” Shutdowns: