Sounds to me like you've pretty much made up your mind to do it. I haven't taken a dog on any 14er, so I don't have perspective on that, but since you asked, I'd say photos 18 and 19 of the route description pretty much tell the story. If you think your dog (presumably with rope) can handle that wall in photo 18, there's nothing else on the route that's harder. Nor is there a great deal of sustained class 3 scrambling to wear you both out - the route is really class 2 most of the way, with a fun but relatively short stretch at 14,000 feet with a lot of scrambling.
The downclimb before the wall, photo 19, is about as tough as the wall itself. There's no person in photo 19 for scale, but know that it's a similar total height to the wall in photo 18. Our group found both the downclimb and the ascent of the wall to be surprisingly easy, for humans, but I don't know about canines. Opposable thumbs are pretty handy in situations like that.
The only other advice I can suggest is to make sure the dog doesn't lock herself into the car before the hike. I read somewhere that kinda sucks.
A vaincre sans periol, on triomphe sans gloire