Here are your stops:
Manchester Center (just because 10-K says so) (2-3 days in)
Rutland / Killington (w/ beer at Inn at Long Trail) taxi there too to Rutland
(6 days from start)
Waitsfield - AP Gap
(in 4-5 more days)
Johnson - walk / hitch into town - everything you need
(4 days more)
17-18 day trip total? aggressive, but totally doable.
if I was going to skip one of these though - just due to spacing, it would be Manchester - sorry 10-K - it is a great town, just too early on a LT hike to need to re-supply there in my opinion