Whether either of these are the same tune, I don't know. The first part of each tune is the same as our mystery tune.

I looked up Tune ID#5098 (Nessa the Mover)

" Titles ? given to this tune in the sources listed below (plus notes of mine about this tune):

"Nessa the Mover (3rd in set The Dingle Berries on Lu 6 and Lu 7) (1st part same as 1st part of Captain Moonlight's Army #3840) (composed by Brendan Larrisey)"

Tune ID#3840 (Captain Moonlight's Army)

"Titles ? given to this tune in the sources listed below (plus notes of mine about this tune):

"Captain Moonlight's Army / John with the Light Brown Hair / Downey / Downey's / Polca Áine / Riding on a Load of Hay / Anne's Polka / West Kerry / Séamus Cussen's (3rd in set Brendan Begley's Polcas on TEdey 1) (also in A Dorian) (1st part same as the 1st parts of Riding on a Load of Hay #6370 and Nessa the Mover #5098, and Ned Connor's #7583 and - not included here - the transcriptions Ceilidh House Polka in book M2 and Gan Ainm in CRE 4 #78; 2nd part same as 2nd part of #6179; also compare 1st part with 1st part of Galway Belle #443 and with 1st part of Riding on a Load of Hay #4297)"

Captain Moonlight's Army - first tune in this set by Slane

Steve, if the tune was played by an Irish band it's not necessarily true that every single tune they played was Irish. Chances are high that it was true, but the folk process tends to include tunes from other countries or cultures as well, and as an example Alan Stivell, Breton harper, included sets of music from Ireland and Scotland on his albums.