I might have the best idea for a "Solo Badge" reward...

Solo Queue. The matchmaker will only match you with other solo players. Simple and effective. This gives players a choice: Are you willing to get matched with premades when u play alone, but be able to queue with friends - or do you truly want solo queue every game?

For how strict the Solo badge requirement is, there still should be plenty of "solo fillers" - it would just be players who also benefit from Dynamic Queue.