Had Custer won, Garryowen would be considered nothing but a flagrant symbol of white conquest of Native Americans.

A few people think of it that way already.

Counterexample: "The Battle Hymn of the Republic."

It's even sung as a hymn in some churches, North and South.

Consider a line like, "May the Hero born of woman crush the serpent with His heel." That means the South. And "Let us die to make men free" is also pretty partisan. Few have a serious problem with the song, however, because it's based on the bible and written in poem-talk.

And nobody in the USA objects to "British Grenadiers," because they don't connect it with 1775 or 1812. Or anything else.