Sorry OP but your home life is a far more pressing issue than your cantankerous DM.
Last year I was very worried about the rising energy costs and didn't put the heating on at all. Money was tight and I was stupidly anxious about the quarterly bill.
I layered up with thermals, thick pyjamas and a hooded dressing gown, a heated throw and a hot water bottle sometimes as well. I thought it was fine. I didn't 'feel' cold. What I didn't realise was that my core was warm, but my hands and feet were not.
Just one instance of getting in a hot bath not realising just how cold my extremities were, and I gave myself chilblains. Which is what your daughter seems to have.
The thing that sticks with me is it took seconds to 'get' chilblains from the bath. But I remember sitting in March trying to work, and my hands and feet were so itchy I couldn't concentrate. I had to go to the pharmacy and get some cream to help the itching, and it still took weeks to get rid of completely.
Now mine was stupid and self inflicted. Your partner has done this to your daughter, which is far worse!
Please, please look after your daughter. At the very least ensure her hands and feet are kept warm (gradually warm them up), and keep up the cream as she already seems to have chilblains. This is neglect, by both you and your partner. If school hear that she has chilblains and she mentions it's because your partner won't allow the heating on, you could be in a lot of trouble.