Firstly, you might be best advised to discuss this with a lawyer who specialises in this area.
There seem to be several issues here.
What is your/the client's definition of a website? The name, the space, the content?
So, the client owns the name. That can be pointed to another web host as they wish, I assume.
Are you offering a tied hosted design which is part of your site?
Is the content bought or leased? Was this made clear at the outset?
Were the limitations of the clients options explained to the client or implied before entering the contract?
I can't offer a way for you to resolve your issue with this client. If it were me and the client can't be made happy to stay, I'd be happy to let them go. Much better, from my point of view, than having an unhappy client.