I'd suggest getting a bike that can take the bigger tires, at least 42-50mm.... and a long wheelbase, like 18". This eliminates cross bikes, and for good reason, they are not intended for your purpose.

A Salsa Fargo seems like it would suit you.

I'd suggest a Rivendell Hunqapillar or Bombadil, but that's beyond your range. The Bombadil is friggin' awesome.

The LHT is Bike Forums favorite cheap bike, this may or may not be a good thing. It would work okay.