My favorite Nashville bar is the Beer Sellar. Great local dive bar downtown with large beer list. They have TVs, though not sure about BTN. We've hit up the Beer Sellar during every Music City Bowl appearance by the Gophers, so maybe they'd be accommodating.
Martin's BBQ downtown. Lots of TV's great food, basically like being at the Ball Park Beer Garden at the state fair. So very outdoorish which is a plus these days.