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It's not "right wing - left wing", it's "right wing - wrong wing".
I realize that doesn't mean that they couldn't make your life miserable for quite a while.
It is a shame, your concept sounds like a fun idea.
Your legal defense is "fair use", as a parodical treatment. Which by no means protects you from being sued. As they have already told you they won't grant permission, I think you'd be well served to write something else, or make your work more generic, not specifically tied by reference to Tolkien's work.
Your book sounds very interesting. As I understand it, the dwarves in The Hobbit and LoTR are an allegory for the way Tolkien perceived plight of the Jews.
I'm dirty and ashamed