Well schools like to do the teaching! I was thinking more of the workload. If the school think you are taking on too much with an extra subject, they might refuse you on that basis. Just in case it affects your other subjects. But, if they agree to it, they probably wouldn't charge you because you wouldn't be a private candidate, you would be entered through that school along side your other subjects. Even though they don't teach that subject, they can still enter you for the exam as one of their students.
Admin costs will be whatever they want to charge you and will depend on the individual establishment. As I said, if you can pursuade your school to enter you, they quite possibly won't charge you.
A lot of school won't take private candidates, whereas colleges tend to be more receptive.