I'd give her a dozen eggs and see what happened. I know your location and the time of year, but there have been pictures on this forum of hens with their chicks in the snow. It is obviously not as good a time of year as you would like, but I think you still have a pretty good chance of success. Obviously you want the chicks. It does depend some on your set-up too. If space is real tight over the winter, it may not be a great idea. My concern would not be the temperatures but how much room Mama has to work with.
If you decide to break her, try putting her in a cage off the ground. The idea is to get air flow under her to cool off her bottom, so a wire bottomed cage works great. Give her food and water, but do not give her anything that looks like a nest. Usually, three days in there is enough for mine, but some can be stubborn.