Although by no-means definitive, you can find some "trivial" tandem statistics here from a survey I conducted last winter: http://www.opinionpower.com/results.cgi?id=149012009
301 - 325 "seemed" to be around the mid-point for tandem team weight (bike weight not included), i.e., under 300 is "lightweight" and over 325 was "heavyweight". However, 370 is by no means in the upper limits of many teams and adding the bike to your body weight certainly drives up the value by 35 - 50lbs, depending on the bike and what you carry when you ride.
All of that said, and having looked at your elevation maps, while I suspect you could ride the event using a generous set of 32mm tires and suffer no ill effects from rapid descents, if you or your stoker is the least bit concerned about speed control I would recommend getting the rear drum, if only for piece of mind UNLESS you can find another tandem team of similar weight and riding style who have completed this tour to querry.
Regarding "other team's impressions & recommendations", always make sure you request some supplemental data such as team weight, experience, like or dislike of speed. Impressions of what's safe, fun, terrifying, etc... are deeply skewed by riders with different predispositions.
Back to your current situation, assuming your tandem has a left-side threaded hub, an Arai drum brake is a lot of added security for not a lot of money. Moreover, if after riding the Tour de Wyo. you decide you don't need it, you can take it off and put it in your spare parts box for a future event if ever needed.