IMO staying at the Keene hostel more sense for those without a car since there's more eating options (and the bus stop is at the Noonmark across the street). [I don't have personal experience with any of the hostels since I live within day hike/climbing range or camp during warmer weather, so I'm not recommending based on any agenda for/against either of the lodging options mentioned, other than ease for the OP] Tmax and Topos is a bit far from Lake Placid/civilization. Fine with a car, not so useful without.
I assume someone in your group is picking you up to bring you to the trailhead for the hike?
I've bummed all over the 'dax without a car before. You just gotta know what's up.
True that, really embrace the dirtbag lifestyle in the high peaks areas and you can get around and live easily. There is even a homeless contingent present in the high peaks...although it might just be the one dude carrying around all his possessions on a sled. Hitchhike from popular trailheads, or busy roads, and there are plenty of friendly people to pick you up.
Great camping, too...if you decide the hostel is a little too ritzy for your budget.
If you're not set on the High Peaks, Pharoah Lake and Mountain is an awesome place to spend a few days. Easy hike in, but there are lean to's all around the lake, all of them beautiful. Pharoah Mountain is a hell of a hike, short, but tough, and there's hiking all around the lake. It's a nice place to take a morning and hike in, set up camp, climb the next day, and take the 3rd day to hike out.