The only problem is one of ambiguity. So long as each language has their own convention to resolve that ambiguity, there's no problem.
Not all languages use the same names for stuff, after all. Abroad, the USA may be called les États-Unis d'Amérique, アメリカ, or a great many other things. And we use our own names for other countries that they don't use for themselves.
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That's just "moon" in another language. If you rename Earth's moon you're just shifting the problem into the Romance languages.
The only problem is one of ambiguity. So long as each language has their own convention to resolve that ambiguity, there's no problem.
Not all languages use the same names for stuff, after all. Abroad, the USA may be called les États-Unis d'Amérique, アメリカ, or a great many other things. And we use our own names for other countries that they don't use for themselves.