The L&C route doesn't go through Yellowstone.

OP: Do some weather research re: places along the route. For example, searching for average daily temperature three forks, montana yields a chart from TWC showing an average daily high in October of 61 and an average nightly low of 32. Again, those are averages over the entire month. It's likely to be below those averages later in the month. Dickinson, ND is 58 and 28 respectively. Only you can decide whether you would be comfortable with those (and possibly worse) conditions, keeping in mind that you could hit some snow on mountain passes.