It does appear that SSPX school is no longer "independent". It has become "public" by accepting funds of the State. Once you do that, the school must adhere to the laws of the State. SSPX must refuse funds and remain "independent".
Here is the USA, we had catholic schools, had. Once the private schools accept "vouchers", they are taking State monies. Catholic schools then must adhere to the laws of the State. They become "Public".
Never take any money from the State or Federal.