I don't quite understand the criteria. Are maps allowed? What about the rest of your normal touring and camping gear? Or are strictly limited to "just a compass and a bike?"
If maps aren't allowed, you ask locals along the way and take notes, as I'm sure Davy Crockett did. My last coast-to-coast ride of 4400 miles took eight weeks. If I had to poke around looking for unknown routes, I can't imagine it taking even twice as long.
I'd be more concerned about crossing the deserts and mountain ranges than urban areas. Cities are relatively easy to skirt.