I tried tweaking the "Sunshine" dog song a little to make it scan more like "You Are My Sunshine.

Here's what I came up with:

To the tune of "Your Are My Sunshine"

Chorus and first verse:
You were our best friend, our very best friend.
You brought us sunshine when skies were grey.
We hope you know, boy, how much we loved you.
Now that sunshine's been taken away.

Verses:

You were a good dog, the very best dog,
And all the time you would wag your tail
You never barked, bit, or even snarled at
The lady carrier who brings us our mail.

We'd brush your coat and we'd scratch your tummy
And pet your velvety pointy ears.
You won our hearts when you were a puppy,
And that love grew through all of those years.

How you loved children! And even kittens!
You loved to play with the boys and girls!
You'd get excited when we came over,
You were the best dog in all of the world.

But now you've gone on to doggie heaven,
And though we're sad, we'll try not to cry,
But one more time we'll say how we loved you
And give you one last "woof-woof" goodbye.

Then (Group howl)