September is beautiful in the high country. You may encounter snow, but it will be gone shortly. Days are warm, high 60's-low 70's maybe,but the nights are cool, and may dip below freezing.
I don't think there would be a problem stump shooting, but I'd check with the DOW just to make sure. You could always get a small game license if that bothers you. Grouse should be in season.
A lot depends on just where you are, and elevation. Most USFS campgrounds will be open to accommodate hunters, so no worries there.
Fishing should be great, then, too.
If you can, drive the highway between Durango and Ouray (US550). Some of the most gorgeous scenery in the world, and I'm not kidding!
Have a great trip.
http://wildlife.state.co.us/