With all the accidents that happen out there, the last thing I want to do is scrape my dog off of the pavement if some cager takes me out. Man's best friend is better off at home, or in a cager with more protection, than on the bike.
Last weekend I saw a tourpack turned into a pet crate. If you do that, then you can pack soft clothes in a bag on top. You may want to consider a second pack and have one for dog and one if you don't want to take it.
I don't ride with my dog, but I was on a ride about a week ago, seen a guy with a medium sized dog, it was gray, about the same color as his beard, that's what cought my eye, anyway he had it on the tank, kind of between his knees.
Terry