The song is by Flanders and Swann and should be in "The Bestiary of Flanders and Swann"; sadly, I lack a copy. If all else fails I will post what I remember after a Decent Interval has elapsed & nobody else has stepped forward. It starts out
A Bradypus, or Sloth am I, I live a life of ease
Contented not to do or die, but idle, as I please
I have three toes on either foot, or half a doz' on both
with leaves and nuts and fruit to eat, how sweet to be a Sloth

BTW, the part I most lack is the fast part, which you seem to have nailed! Between the two of us we can reconstruct it.