I don't have experience backpacking in winter but I was born in North Dakota and lived there for my first 28 years.
If I can be bold enough to offer some advice, then here it is.
First, be absolutely sure that you have enough clothing and insulation to take care of yourself. The Badlands tend to be warmer than the rest of the state, and they offer lots of little nooks and gullies full of brush where a person can shelter from the wind, but winters can be fierce. I would definitely not recommend going as light as possible because it may mean trying to go lighter than possible. Once the weather changes there is nothing you can do about it.
The winter of 1887-1888 killed off most of the cattle business in that part of Dakota Territory because it killed off most of the cattle. Hundreds if not thousands of cattle froze to death. You can have a blizzard arise without much warning and even on nice days it can be windy. Temperatures of -25 degrees with a 25 or 30 mile per hour wind are not too unusual and suck all the heat from a body in minutes. Days like that are unremittingly evil, and you cannot get away from the wind.
I have been out a couple of times at -40 degrees (when it's always been calm) and can tell you that it is different in kind more than in degree (it's not a colder kind of -25 degree day at all). It's very unlikely that you'll see weather like that, but it is possible. February tends to be the coldest month. During my last winter in the state I worked outside. My coldest day was -35 degrees in the morning, rising to -15 degrees by about 1:00 p.m., and I could duck inside the office any time to warm up. It was interesting, but I wasn't out on the plains with a candle and a blanket.
Hiking hard in relatively calm weather is OK. You can even go without a jacket sometimes, especially if the day is sunny, but wind can kill you dead really fast, or remove fingers, ears, or a nose due to frostbite. Every year some people get caught, in stranded cars, and freeze to death while sitting on the highway somewhere. You will be out in the open.
I would also recommend knowing where each ranch house is along the route so you can bail out or at least find a few hours of shelter if needed. I expect that most people would be delighted to harbor a couple of interesting lunatics for a while. They tend to be very friendly folks and like company.
Water is likely to be a severe problem if there is little snow. North Dakota is a semi desert, and most precipitation falls in May, June, and July. The Badlands are in the driest part of the state where annual precipitation hangs around 10 to 15 inches, or less. It might be possible to cache some water in small chunks (like jars full of ice cubes) so it's easier to thaw as needed.
I've been on the Maah Daah Hey twice in summer, and loved it, and want to go back, and have even thought about a winter trip. It could be great if the weather isn't terrible and you are prepared. Please think six or eight times about how much gear you will could really need.