Having been enticed back to Melnet after a long absence, I've explored a few threads that looked interesting, and came across this one. I'm impressed that Jack Campin guessed correctly that 'Mr Moore' began life as a G minor tune, since that is precisely the key in which Thomas Watts noted it 200-odd years ago.
Actually I made another tweak additional to the key change to E minor: in the MS, the tune has a key signature of one flat, which means the E notes are naturals, and the tune is in G Dorian. I tried it first in E Dorian, and found the sound of the C#s really grated, so I changed them all to C naturals, thus making the tune Aeolian. It's funny, because I usually love the Dorian mode - especially in song melodies - but this one didn't work for me. But do try it for yourselves and see what you think.
I don't know whether Jack has an opinion on this, but one fiddler suggested to me that the tune might actually have been Aeolian, but that lazy fiddlers found the open E string an easier option than fingering Eb!