Can't help on locating a recording, (if all else fails, maybe you could get a group of friends together to sing it & record them?) but responding to Susan's post, which I missed the first time round...
I Had a Little Dog (named Do-fi) - Bingo??

No... very different from Bingo. Doesn't seem to be in the DT or in the Forum, so here 'tis:

I had a little dog named Do-fi* (*think Fido pronounced backwards... sort of!)
I had him ever since he puzz a wupp
I taught him how to sit upon his lind hegs
And to hold his punt fraws up
And tag his wail...