As soon as I wrote that I had a vague memory resurface of being able to amend the ‘system’ flag for the badge in the database using the rails console. Let me give it a run through and see if I’m remembering right. 
Yes, I was remembering right. If you change the ‘system’ field for the badge from true to false using a rails command you can then edit the default badge in the UI [1]
Extra details
I used:
Badge.where(id:6).update(system:false)
Though be cautious when doing anything in the rails console as accidents can happen. 
(at your own risk, of course
) ↩︎