Hard to beat Old Dog Trey for a really good dog song.
There's also White Cat and Black Cat, but folks may get weird about the "shave him to the tail" part. Then again, maybe not.
On the Appalachian dulcimer NING site, we've had a couple of threads naming bunches of songs that include both dogs and cats:
Many of the songs mentioned are old and traditional, and there might well be a couple of 1800's songs mixed in, but you'd have to dig through the posts on both threads and maybe look some up to see their earliest published dates. Might jog someone's memory in some helpful way, though!
Old Rattler, http://www.youtube.com/watch?v=d5BzRYME1ww
The Hound Dog Song, http://www.youtube.com/watch?v=hJ3rXbFn8qE&feature=related Recorded in 1926, this may date back further. Not sure if any of these are from the 1800's.
Can't forget this
Der Deitcher's Dog
Try these sites, if not already familiar with them: