You said "She picked it up from an Irish folk band back in the 80s and has never known the title."
When you say and Irish folk band was the band playing actually Irish people, playing in Ireland, or somewhere else, or was it a band playing Irish folk, not necessarily with Irish-born band members? And if it was the latter, where were they playing, country, venue, any further information? And can your wife remember the name of the band or any other songs or tunes that they played? Any further information could be useful.
The tune Riding On A Load Of Hay seems very close to the MIDI tune you gave us, in my opinion. Ignore the timing for the sake of the quest and just listen to the notes of the tune.