My teacher said that this idiom would never be told by an American and is British English. What is an alternative way to say this in American English?
I believe your teacher is wrong. I am an American and "What more could one ask for?" sounds perfectly natural, if a bit formal. This is not one of those idioms that comes in one form that everyone uses consistently. You can substitute "anyone" or "you" or "a person" (for instance) in place of "one", and you could substitute "want" or "wish for" (again, just as examples) instead of "ask for".
I think its the "one" construction that would be highly unusual in colloquial American English. Rephrase it without that and you are probably OK. For example, I'm pretty sure I've heard "What more could anybody ask for?"
I've also heard @convoke's "What more could you want?", as well as "What more could you/anyone need?"