Is there a module for automatic menu translation?
Short answer: no

Long answer: I wouldn't recommend to try to 'automatically translate' any of your content – especially not translating the navigation while keeping the content in the original language. You will probably rather quickly have lots of frustrated users (or no users at all) that don't find content in their language, even when they are 'clicking the right link'. 

Additionally you will probably be penalized by search engines for duplicate content.

Maybe you should ask over at the ux forum what people think of the idea? I suppose hardly anybody would recommend to have that type of mis-leading navigation structure on a website.

There are however various modules to tranlate menues (and fields and taxonomies etc.) on the fly, while you are creating new content items – but that you probably already know – and that's probably not what you were looking for.