People have done this kind of touring with almost any kind of bike, so there's a good chance you can make it work for you. That being said! Many of us prefer lower gears (I hope yours is the triple version); if you can change out the cassette for an 11-34, and perhaps switch the derailer to match, that'll get you close. Pack light, you've got to climb 100,000 feet or more, and pull whatever you pack up those hills. Finally, I'd suggest you change your pedals to mountain bike pedals and shoes, so you can walk (and push) if necessary.
Are you planning to camp and cook? Your budget looks slightly low to me, but you might be able to make it if you don't eat out or stay in a motel very often.