While Im quite partial to the tune of Battle Hymn, it doesent really strike me as an "anthem" song, you cant really see it being sung by 10,000 people at a baseball game, and more importantly, the lyrics have nothing to do with America, her history or her people. If you didnt already know the history of the hymn, you would have no idea it was about America.
America the Beautiful is truely awe inspiring, so I would go with that one, and Honestly Star Spangled is a great song, It would be nice however, if people sung it more as a choir at official events
Yankee Doodle is also a good one, despite some... questionable lyrics, and it being apparently really hard to play for bands