I think you might have to write the songs, probably using the tunes of selected carols.   

For example, the tune of Good King Wenceslas was probably around at the time (used for "Tempus Adest Floridum", which is in Carmina Burana). It could make quite a good tune for getting marchers to step it out a bit.

And there are other carol tunes dating back to around then. And in fact the Agincourt song was generally referred to as the Agincourt Carol.