Which trails exactly does this encompass?
Looking at the map I think you're talking about the Black Mountain Crest Trail down to Mt. Mitchell, and then from there I'm uncertain as to your path.. Are you finishing up with Heartbreak Ridge Trail/Star Gap? If so which route are you connecting through?
I hiked that a couple of weeks ago from the south end at Graphite up to the terminus at the BRP (and a couple miles up the BRP), and there's very little water until the bottom @ Graphite there. That being said, even with the ~3,000ft elevation change (and S to N is the uphill path), plus the 1-2miles on the BRP my dog and I made it in about 3.5 hours, with atleast one 10-15 min break.
Those trails combined are listed in the HToNC as strenuous however, and my average tends to be in the 14-16mpd on short trips.
The Black Mountain Crest trail is 12mi's with 3,500ft of elevation change, and described as strenuous also, but I haven't checked that out yet.
We're already having nights in the 40's, so it could swing either way, definitely count on freezing temps at elevation, and I'd be prepared for mid-20's personally JIC.
I'm not too familiar yet with alot of the area trails, as we just moved here, but we're about 20 mins from the southern section, and not much farther by car when the BRP is open. If you need any help with taxi, let me know.