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No, at the time it was ratified, the entire BOR was intended to apply only to the federal government. That's clear from every statement made at the time from pretty much everyone on every side of the issue.

Is that so? Then what did they mean by Article VI?

This Constitution, and the laws of the United States which shall be made in pursuance thereof; and all treaties made, or which shall be made, under the authority of the United States, shall be the supreme law of the land; and the judges in every state shall be bound thereby, anything in the Constitution or laws of any State to the contrary notwithstanding.
The Senators and Representatives before mentioned, and the members of the several state legislatures, and all executive and judicial officers, both of the United States and of the several states, shall be bound by oath or affirmation, to support this Constitution; but no religious test shall ever be required as a qualification to any office or public trust under the United States.

They meant that in the event of a conflict between state law and the Constitution, the Constitution prevails. But in order for a such a conflict to occur, the Constitution would actually have to apply to the situation that the state law applies to. If a particular provision of the Constitution only applies to the federal government to begin with, then there's no way a state law would conflict with that.