Not NWT, but I hunted the Yukon late Sept./early Oct. Cold and had lots of snow that year, like 18 inches to 'wade' through. Thankfully we had horses to use.
By that time most outfitters are focused on caribou and moose. Some hunts would be available but ususually snow will be pushing sheep around or down a bit . Fixed wing extraction is risky with chance of snow. not so if you are with a heli outfit.
I wouldn’t go on a designated sheep hunt in the NWT during mid to late September. You could possibly add on a sheep on a trophy fee basis on a moose hunt during that time of year . You could end up with Indian summer. I spent nearly a month in the Mackenzie Mtns Sept 10 - Oct 5th time frame. My guide and I were stranded 5 days on the side of a mountain sheep hunting, waiting to be extracted by a helicopter due to a blizzard, helicopter could not fly to get us out with two failed attempts due to the blizzard. Later in the hunt we were once again socked in a spike cabin with 3 other hunters and their guides for another 4 days due to another big snow storm and neither choppers or planes could fly. We ran out of groceries except for 2 moose, a caribou and a couple sheep. Again flying out of the spike cabin on the end of the hunt we flew out by fixed wing float plane we experienced more bad weather and the plane iced up so bad I wasn’t sure we would make it. The pilot was scraping the ice off windshield of the plane to be able to see to fly. Be prepared for severe weather during that time of the year in the Mackenzie mountains.