I will do that again without typos and silly interference from the spellchecker:

It's not useful to obsess about 'authenticity' of the music played in a series like that. I very much doubt anyone going for 'an auld Oirish tune, from Ireland' would have come up with Garryowen. That one is a wholly American obsession in the wake of Custer and all that.

'I dreamt that I dwelt in marble halls' you asked after in another post is from the 1843 opera 'the Bohemian girl' by Balfe. That doesn't fit the timeline very well. But does it matter?