My encyclopedia says that Thomas Moore (the author of Bendemeer's Stream) wrote many poems to be sung to traditional Irish airs. Thus the tune to the song we call Bendemeer's Stream was in existence before the words. Possibly it was called "Portadown".

Surely if you are changing the text from English to Esperanto, you will have to change the meter to fit, so why not just go with a modified version of the melody to Bendemeer's Stream? People will like it.