If I were doing it all over again:
Traditional mountain bike
Bikepacking setup, or at least a good frame bag
2.4"-2.5" tires
Front suspension only
Possibly a suspension seatpost or sprung saddle
Possibly tubeless tires (I need to learn more)
That said, I'm convinced you could do the whole thing on a cyclocross bike with 35 mm tires if you were determined, you packed light, you didn't mind walking on 5-10 occasions. Most of the time, that setup wouldn't be a problem at all.
Washboard and sand are your worst enemies, if that tells you anything about what to bring/expect.
There are alternative routes, mostly on pavement, around some of the worst areas. Most notably in Montana and New Mexico.