I think 6/8 makes decent sense as it preserves the 2-beats per measure while also allowing for 3-beat subdivisions. Here's a rough working interpretation of how it would go.

  Oh,       SHEN an DOE--, I LOVE your DAUGH ter
  (pickup)  1    3  4      6 1     3   4     6
  A a WAY, you ROLL-ing RI-ver
  1 3 4    6   1    3   4  6

I put this tune in MuseScore and it didn't sound awful but it is a bit jumpy. Here's how it sounds for anyone who has a hard time following that: https://youtu.be/OQfNaZ2hzKY

(Ignore the ungodly abomination that is the [lack of] key signature in that transcription, I was mostly focused on the tune)