As expected, there are many variations of these lyrics. I learned mine from the Royal Air Force Attache to the UK Embassy in Seoul when I was a fighter pilot in the USAF in the 1980s and he dropped in on our Officer's Club at Osan Air Base. In his version, the last line went:

So if your dog's in great pain each time he takes a shit,
There is a simple reason: his asshole doesn't fit!
And this is why, when dogs meet, each one will drop his bone,
To sniff the other's asshole and see if it's his own.