When did they move Burlington to southern VT? When I was there again last September it was still about 40 miles from the Canadian border. Or did you mean Brattleboro?
I have ridden south from either Brattleboro, or Burlington through Brattleboro, to join ACA's Atlantic Coast Route during trips home to Philly. Twice in N. Canaan, CT and once in Millerton, NY. As noted, campgrounds start shutting down relatively early. For example, Lake Bomoseen S.P. near Fair Haven, VT, shuts down after Labor Day Weekend. Nearby Half Moon Pond S.P. closes mid-October. During my trip starting last September, the place in the southern Berkshires I camped at was closing the following weekend. ("Winter comes early around here." the owner remarked.) Lone Oak, a few miles east of N. Canaan, CT, closes mid-October. The Rudd Pond camping area of Taconic State Park, a bit off route from Millerton, NY, closes after Labor Day Weekend. You will pretty much find the same as you get into NJ and PA, at least with respect to private campgrounds. Worthington State Forest in NJ stays open year 'round, but they shut off the water at some point. And I don't think I would want to risk sub-freezing temperatures at the time you're talking. I have ridden that area in mid-October when nights have been around freezing. Dogwood Haven, just off route in Upper Black Eddy, PA, closes at the end of October.
Luckily for you, whether you are in Burlington or Brattleboro, you can take your bike on Amtrak's Vermonter as far south as D.C. No boxing of the bike required. You just need to reserve one of the 3 bike spaces. At the time of year you are looking at, that probably will not be a problem.