WordPress already does all this by default. There is not reason for you to need to modify the slugs work, you just need to configure your custom taxonomy to have the rewrite slug ‘fine-art’.
I think you can then just create your page with that same ‘fine-art’ slug. I believe it should still find that page regardless of the taxonomy name having the same slug. Then all you need to do in your template is what you appear to already be doing, which is generate the links for the terms you want to show.
If this still is not working, it might be helpful for you to include your custom taxonomy declaration.
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