I had a prof in college who taught a course on US Government. I remember him talking about a Constitutional Convention and wondering, once called and in session, how it would operate? Could you have a Convention that was limited to what it could act on or one where it could pass on anything it wished? Would the State Legislatures have to vote on all proposed changes or could they cherry pick amendments?
The Convention would be for the purpose of proposing amendments. I suppose they could propose whatever amendments they want, but they cannot wholesale write a new Constitution. That said, I suppose they could try to do what the Senate does with House bills and pass a single amendment that says "Strike out the entire Constitution and replace it with this..." I doubt that an amendment proposal like that would win a vote at the convention, and even if it does, I'd expect 13 states to reject it.
The states wouldn't cherry-pick amendments, they would vote on each separately, yea or nay. The current Bill Of Rights were voted on separately, with different combinations of states voting for different amendments.
-PJ