No problem.
Since you can assign badges to categories, all you need to do is create new categories, one per type of badge. Then you assign these categories to as many, or as few, or only one product - as you wish - in addition to the "regular" categories you have assigned to them.
If you run into trouble with category or product listings (because you don't want these new categories to show, or products to be listed twice), unpublish the new badged categories, or place them outside the "Product category" in "Root".