July 1 - you will definitely encounter snow in segment 7, up near the top of the big climb. That is supposedly the last spot on the CT to melt out. It was still there on July 25 of this past summer when we went over. The same is also true between Searle and Kokomo Passes in segment 8.
A fair number of folks slack-pack segment 7 from Breck to Copper Mtn. Sure wish we had done that! Leave your gear at the Fireside Inn B&B in Breckenridge (they're used to this and even advocate it), take the free Summit Stage bus to the trailhead and day hike over to Copper. Pick up the Summit Stage again at the Passage Point stop at Copper, ride it back to Frisco, and change to the Summit Stage bus back to Breck, Stay overnight again, then reverse this back to Passage Point. Walk through Coppr Mtn to the Plaza and take the gravel path back up to the CT. Turn right and head for Durango.
Edit: the map is too small to read well, but the red line is the CT. The red dotted line is a 50 yard gravel path from the Plaza right back to the CT. Get off at Passage Point, walk thru the tunnel and back to the Plaza. Stop at the outfitter if needed, then hop on the path and walk on.
As a CO resident, if you have a friend with a car who could meet you at Tennessee Pass, you could also slack pack segment 8. It would be a long day, but you'd save humping your pack over those two huge climbs and could spend one of the nights at the B&B and the next at the Leadville Hostel. Consider that in lieu of (or as an addition to) a zero day in Breck.