If you can find blank covered buttons in the same size, you could then take it apart without worrying about destroying it. Also, is there an area inside the coat (like a hem area) that you could take some fabric from then mend? (In case you weren't able to repair the little circle of fabric).

Well, if nothing else helps move that button to the bottom of the coat. Only you will see the imperfection !! Can/would you replace all the buttons? I would try to fix the one you have by maybe patching it like zone suggests.

It sounds like they used too small a piece of fabric to begin with. If they couldn't get it to work, you might not be able to, either. I would move it, too, like Salena said, and perhaps take a dab of acrylic paint to cover up the metal that shows.

If the coat was purchased new, did you look inside for an extra button? If there isn't, then I'd ask the store for a replacement button. Or write to the maker of the coat, they may help you.