Well, it all depends on how much shelter you really want. Single-pole tarp/tent hybrids such as the Betamid offer the advantage of lower weight (it tips the scales at a mere 2 pounds, 2 ounces) in return for a single-wall, non-breathable design, and of course, no floor. In the right hands, a Betamid can be set up in the snow to make a sort of semi-igloo, and can be a perfectly adequate winter tent. You’ll need to pack along some sort of tarp for a floor, however (well, that’s optional, but I think it would be useful), and even at that you’ll find that snow and moisture can find a variety of ways to creep in. You’d certainly survive, and might not even miss the sealed floor. Then again, you could end up wet, cold, and miserable.
On the other hand, I’m not sure how much better you’ll do in the sub-$200 tent market. Not a lot out there in the four-season realm, particularly if you want to house as many as three people. Probably your best option is Eureka’s Alpenlite 2XT, which is a bona fide all-weather tent for a mere $200. It weighs a very impressively light six pounds, but it’s a little small - adequate for two people, impossible for three. From there the price tag ramps up pretty quickly. Walrus’s Starstreme SE ($279), although technically a three-season tent, would manage well in the winter and can sleep three if need be. And there’s REI’s Convert Mountain ($249), a well-designed little tent with a sort of tunnel entrance that helps keep out blowing snow. It’s primarily for two people, however. So there are a few choices, at least. My advice is to ditch the third member of the party, and get the Alpenlite.