@billcsho - You’re wrong. Did you read Publ. 970? The OP can do exactly as he’s proposing, and Publ. 970 makes this clear.
You are correct about one thing, though - there is indeed no reason to give a tax credit to a student who’s already receiving a full ride. But that’s beside the point. Clearly, that wasn’t the purpose of the AOTC, but those students who receive full ride scholarships are so few as to be statistically insignificant. They are simply unintended beneficiaries of the law . . . and there’s no reason for them not to take advantage of it!