Quote - I want to make some mountains look like they are miles away, but I don't want to have to drag them miles away from the camera.

 
The only natural way to do it is to drag them 'miles' away from the camara, Bryce's fog looks so much more natural than adding cloud planes, although it is still a good idea, as an addition.
What I do to get a decent distance in Bryce, is to create a scene, when i'm happy with it, expand the entire scene maybe 4 times or more from what it was. This gives great levels of distance fog.