Could I demand that my child be enrolled in another class?

You can always ask but you should have several reasons for the request. Since it is a private school and your child is only three years old so it is not likely you can claim the teacher is not competent academically to teach, I do not see how you would have any right to insist on any change. Even if it were a public school, you probably would have no right to insist on a change.

The answer depends on any written agreement that may exist between the school and you, and any written policies and procedures adopted by the school. I think these are the only sources from which the right you seek to claim might arise. Good Luck.

I guess that if you are paying, you can do what you want.

You should have a contract with the school which should answer that question. Sometimes the school's policies and procedures will act as the contents of the contract. If such is the case with your school, check them out for the answer. Additionally, you could always ask the head master. Good luck.