On my board, instead of making them into groups, we showed badges in a custom profile field.
So, you would go to the acp -> profile fields, abd create a new field ("backer", I presume?) And then in the display, link to the image. The badge will show up under their primary group icon on their sidepanel.
You can also member search by profile field, so for us, there was no need to make a secondary group, or install a plugin for us to keep track.