Public transportation, no, not that I know about. However, you might get CO Mtn Express to do it, but it will cost you.
Contact CO Mountain Express (http://ridecme.com or 1-800-525-6363). They run several shuttle vans to and from Denver airport to Frisco, Vail, Breck, etc. They might do Kenosha Pass for you. I've taken them from the Fireside Inn to DIA a couple of times.
The Denver trailhead(s) are a bear to get to. There's no public transport. The only bus that went near the place has been discontinued. You could park your car there (if that's allowed) and walk to Breck, then take CME back to Denver, but you'll probably have to take a cab to your car. Again, ask CME what they can do for you.
Looking at your gear list, I don't see any bandanas. You'll need something for a neck cape to keep the sun off you, especially in the big burn area at the start of segment 2. You'll also want sunscreen. Frankly, I'd ditch the BB cap for a wide brimmed hat; gives both sun and rain protection far better than a BB cap.
You will also want a ground sheet of some sort. Your shelter has a netting floor, and your NeoAire is vulnerable to sharp objects on the ground. Put the ground cover (Tyvek, polycryo, etc. on top of the netting floor under your pad.
Otherwise, it looks ok on the 1st pass.
For some truly targeted info, read the CT forum at Trailforums.com A posting there for transportation help could surface a local resident to help you.