Early September is when you start getting a few short storms in Yellowstone (although it can snow there any time). The weather in the central CO Rockies starts a little later, you can always get a thunderstorm or hail storm, but mid September you should be OK. By late September / early October, storms that last a few days are more common, and you should be prepared to wait them out, preferably indoors. "not afraid of a few snowflakes" may apply on the plains, but doesn't cover 20-mile descents in hail, freezing rain, 0 visibility - hypothermia is a real possibility.
Don't rule out the Lewis and Clark to around Bozeman, then south through Yellowstone and on to the Transam. L&C is awesome. (I haven't done the NT or WE, so can't compare). Also, don't rule out a make-your-own connector route, by the time you get there you'll know how.
A few years ago I did L&C to my own mid/northern route, starting August 1, finished in New Jersey Oct 13. The weather was great, but I would not have wanted to be much later - I had snow & hail in Yellowstone, otherwise great weather, but the days were getting short and cold by the end.
Notice on the WE that you do still have to go through high mountains - St. George, Telluride and Monarch Pass are ski areas! So you want to pay attention to when you will be around there - heading south onto the WE is not going to get you around the south side of the mountains, it just gets you to the south *end* of the mountains. Late September would be about the latest I would want to be getting out of the mountains into Pueblo.
My gut feel, FWIW, YMMV, $0.02 - if you aren't heading East by mid-august, you should head all the way south onto the ST.
Hope this helps - your trip sounds fantastic.