If I had to guess, figure on getting across Idaho by the end of September,
Please you would have to explain a little further re Idaho etc as im a complete newbie to touring especially in the US as i live in London.
Being in London, you do not know what mountains are. Maybe you have visited the Alps in the summer. Idaho has the Bitterroot mountains. 100 km east are the Rocky Mountains. 200 km west are the Sierra Cascades. Its very easy to have snow in these mountains in late September. Usually early September is fine. Usually. Imagine suggesting to someone from the very far north of Scotland, on the North Sea, about going on a bike tour in late September. I'd guess that person would beat your head with a hammer for saying something so stupid. Riding around the Rockies and Bitterroot mountains in late September is about the same thing.