I think I found it! I posted a message to the Papillon discussion page, PAP-L, and got several responses.

It appears that the movie is "Heaven and Earth" with Tommy Lee Jones. In the movie, the dogs belong to his mother and he describes them as "the butt-ugliest dogs in the world." Now I am really interested because the movie is not of the type I expected and I am dying to see what an ugly papillon looks like!

I was also informed that there was a papillon in the original Glen Close 101 Dalmations and possibly in the music video "Who Let The Dogs Out." Any opinions?