Cross bike might work, but...
We did the WY and CO section of the Great Divide Trail in 2002. On hardtails with BoB trailers. That seemed to be an appropriate set up. The one other person we me on the trail that year (it was exceptionally dry and there were loads of forest fires across CO) did it on a Ti frame cross bike, a Carpe Diem with a carbon, if I recall correctly, using just two panniers on the back. We traveled together for a few days, but his set up was quite a bit faster; but his wrists suffered even with 38 tires. So, a CX can of course do it. You choose the level of comfort (comfort being a relatively concept, you suffer either way!).
Enjoy the route - and take the minimum amount of stuff with you.