September can provide wonderful hiking conditions in Colorado. You must be prepared, however, for the possible snowstorm. (Mid-September last year I got hit by a near white-out near the CT/CDT intersection east of Silverton.) October just brings more of everything - more cold temps and more storms. I’ll defer to anyone who has actually hiked the CT in October, but I suspect you’ll end up spending a lot of time in towns waiting out bad weather. Given that fact, I don’t know if you would complete a thru-hike before the trail become altogether impassable (unless you can bang out steady 20+mile days).
I hope someone with more late-fall experience on the CT has better news for you as I hate to rain on a good adventure! Happy Trails.
Matt