I am also a Home carer and our company had the same dilemma. In the end, they paid double for xmas/boxing day and bank holiday rates on the Monday/Tuesday. The difficulty was that the Council refused to pay them extra for xmas/Boxing Day as they were not Bank Holidays. However, they had to pay extra to staff, because it is stated in our Terms and Conditions.
I would get a copy of yours and see exactly what you should be getting. However, I think you do have to pay back any overpayment:-(