I have ridden across the "Kanc" at least a dozen times. Yes there is often ongoing construction, and you may also end up competing with the camper-towing cars for road space. Best times are usually mid-week as the weekends are notorious for tourist traffic. And Fall foliage season can be hectic as well. Late September through mid October for that. A couple of times I'd started at Loon Mt ski area on the western end, and ridden a century ride, going up through Franconia Notch via the bike path (required because of I-93 being off limits to cyclists), then across to Twin Mountain, down Rt 302 and Rt 16 to Conway and back across the Kanc. You can shorten that loop a bit by taking Bear Notch Rd from Bartlett and intersecting the Kanc around 10 miles from the eastern end. That cuts out cycling through the busy center of North Conway and makes the ride around 85 miles or so. There are campgrounds (state run) on the Kanc in about 4 places that are usually crowded on the weekends but often have available space during mid-week. First come, first served basis, no reservations.