The trick with two-wheeled trailers would be to avoid the single track--a bummer for mtn. bikers, but there isn't much on the GDMBR anyway. The other issue would be some of the rough roads--especially washboard. Get a good Burley (we tour on pavement with a "Solo" that carries our border collie/blue heeler) and see how it goes: careful practice runs on dirt roads. PLAN on being slower than someone without a dog--especially on the down hills. I think it can be done (and I'd love to try it) but practice and patience will be key. Keep in mind, too, the extra weight of food, water, etc. There used to be a photo on the Burley site of a woman who took TWO dogs (looked like border collies) on the Great Divide for about 800 miles! So it can be done. Practice, boldness, and a clear, calm head for the extra work and care required would be requirements.
Give 'er a go! Report back.