Good question. As Dr. D suggests, I believe the range has been traversed on skis before. Any question like this is hard to get a handle on without descending into Clinton-esque (and thereby Presidential) analysis of the meanings of "traverse" and "on skis". Do you have to go over every named summit to "traverse" the range? How much "taking skis off and hiking" is permissible on the ascent? What about the descent? For that matter, what does "skiable" mean? If you are wearing skis but are carefully sidestepping across the felsenmeer, are you skiing? As a seeker of adventure (as opposed to a universally-accepted record), perhaps one's own judgment is required.
Meta-analysis aside, I consider it a doable itinerary, especially if you stay low. (I actually think it is likely doable entirely on skis, including all the summits, but that's pure stunt and not really skiing.) For example, I bet one could take Valley Way, along to Gulfside, to the summit of Washington, down to Crawford Path.
Despite the alpine rigors of the route, I think it would be easier on lighter skis than on heavier. Unless you deviate from the route, you won't be bombing down deep, open snowfields at high speed. You will be climbing a steep, narrow trail to get above treeline; you will then traverse over a dozen miles of broken, windswept plateau, over a mix of icy scree and sastrugi, possibly climbing talus and scree to summits, all of which will likely occur with a strong wind across your face, before you descend yet another steep narrow trail. I've skied portions of the route on a variety of gear, but have found light touring gear suits my style better than heavy downhill-oriented gear for this area. (Obvious exceptions if you plan to descend any of the principal drainages, as opposed to sticking to the ridge.)
Also worth noting is last spring's
TR: Circumnavigation of Mount Washington, NH 4.16.09 in which the members of the party "ski all 6 major drainages of Mount Washington in a single day". Something of the opposite interpretation of a ski traverse in that it focuses on steep terrain as opposed to a longitudinal traverse, but impressive.