I think your answer is not grammatically incorrect, and that FumbleFingers intuition is sound. I analyze it this way:
The standard paradigm for what has been called here the "third conditional" is a counterfactual statement about two past events:
"If I had studied, I would have passed the test."
That paradigm can be extended to a past state of affairs that affected a past event:
"If I had been taller, I could have ridden the roller-coaster."
The excercise you give, however, has two twists to it, in that the conditional clause poses a counterfactual about an on-going state of affairs beginning in the past but extending to the present, and does it in the second person. This makes the application of rules about the sequence of tenses very ambiguous, and it becomes less natural to follow the usual rule for a third conditional. To see this, imagine addressing your neighbor's child, who is not as good a pianist as your child, as follows: "If you were my child, I would have gotten you a better piano teacher." It would be somewhat unnatural to say instead: "If you had been my child, I would have gotten you [etc]," because fixing the counterfctual in the past could be taken to suggest that the counterfactual subsequently stopped being a counterfactual. Thinking about it, I think I could be persuaded with very little effort that your answer is actually better.
That being said, I agree with most of the other comments here that for purposes of a standardized educational exercise, the book answer would be the conventional one.