I'm a road tourist, no info on the divide, but I've done some of the GP. Last summer I rode from Wyoming to Jasper and back. I left Jasper 8/14 and rode to Yellowstone, had great weather in Canada, a little rain in Yellowstone, missed snow by a couple of days, but there were still some warm days after, I think. It was just starting to feel like it was getting on the late side - 9/3 in Yellowstone. I wasn't setting any speed records. I did have some rain on the Icefields Parkway on the way north, sat out a day or two, but it wasn't very cold.
My gut feel is, unless you are really fast, you'll be OK until around Walden (south of Rawlins) and then south of there you head up into the high country in Colorado. September is the shoulder season - could be nice, could be stormy. You could always bail out of the mountains to the east. Mid to late september it's not a surprise to get snow & hail up high, but the front range should still be really nice. Or, you could head over to Utah, September/October around Moab is really good.
My impression is the divide is generally higher than the roads, so probably on the too-late side.
Don't know if you want suggestions, but that would be an absolutely lovely time to ride the pacific coast, if you are looking for an alternative.