The Ballad of Ira Hayes
Just heard this on the radio today. Apparently, though covered a lot at the time, this fell off the radar over the decades.
https://en.wikipedia.org/wiki/The_Ballad_of_Ira_Hayes
It's quite decent, and the story (downtrodden, volunteers for war, celebrated, forgotten) tracks pretty well with the overall movie themes.
The version here (there's a 30 second preview available for free) I think would have fit rather neatly into something like the closing credits, for example:
http://www.amazon.com/The-Ballad-of-Ira-Hayes/dp/B00LBQNC1Q