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Assuming you aren't going to fill and repaint, most hardware stores sell small, flat, black plastic plugs in their misc. screw and bolt section. They come in various diameters, and a 1/4 inch size should be almost invisible on your black vehicle. I've also used a paper punch to make some black dots from vinyl electrical tape (or similar stick material) that can cover the holes nicely.
My Clubman S came off with fish line...no holes. I can't speak for MINI logo badge, though. The plastic plugs look okay...better than holes, not as good as bodywork/paint.