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This is a complicated situation, and please understand that all I am telling you is just basic legal information. I cannot provide you with legal advice. Generally, it would not make sense to have two separate entities contracting with the organization for the same work. It does not make any sense for the board to agree to contract with her corporation and not her if some people do not wish to do that. It certainly would not help your organization to have a contract with a corporation because if that corporation has no assets and something goes wrong no one is going to be on the hook for damages.
If this person is asking for too much or this is too complicated it seems that down the road you may all be asking for trouble given that she is making a very simple situation much more complicated than it need be.
I hope that helps as a starting point.
This is supposed to be a conversation. We can go up and back as much as you like but given your approach I prefer to opt out.