i just got back from a week out of town, otherwise i would've recommended that you not try this, due to the snow that would be on the trail, at least at higher elevations and in the trees / north-facing areas.
i tried to do the TRT in september in five days (ca. 33 miles/day), and bailed after 3.5 days and 110 miles or so due to a bruised foot that was making it pretty uncomfortable. water then was a big deal on the E and NE (as always). so a couple of months later, with some snow, and another 10 mpd or so, sounds pretty darn difficult (without lots of trail running, route finding, and hiking in the dark).
anyway, there's always next year after the snows melt!