We sort of have the same issue with an older home in Wisconsin. DH has been here 35 years and once he came home and found that something went out causing the water to run the holding tank dry forcing the pump to run constantly. You run the risk of either flooding the basement or burning out your water pump....and that can happen anytime of year. Winter time only adds to the potential issues.
I know a couple who had a home on the market in the winter and one bedroom got too cold and the pipes froze in that room and messed the whole works.
I think the best thing to do would be to have a neighbor check daily.