I done a few short sections of the route, and while I'm sure a CX could do it, I think a hardtail mountain bike is a much better choice. It's rough enough you need the front suspension, you want the wide wheels for sand and loose stuff, and you need the triple in the front for climbing.
Maybe something mostercrossish (wider tires than a typical cross bike, and often a triple crank or at least lower gearing). People have done the GDBR on singlespeeds so anything is possible, and some of the bikes did look very crossish. Dirt drops seem popular for the hand positions.
That said I think that the most recent race was won on a carbon hardtail. But that's racing.
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.
Rick Hunter started the great divide race on one of his cross bikes. He dropped out at some point. Maybe he wrote about the bike choice somewhere. The vast majority on last years race had a hardtail 29er and most had suspension. The winner had a cool setup with mtb bars and (IIRC) a second stem to give more handlebar space and a mount for some aerobars.