I know you have probably been doing this but you need to kill a lot of scorn. The public event nearly alway drops one for me.
This is a known bug, see https://www.bungie.net/en/Forums/Post/252323677?sort=0&page=0 , I have killed 1000's of scorn inc 100's if not 1000s of yellow bars, done public, lost sectors etc, spent well in excess of 20 hours on this part of the task alone and still nothing, its a joke.
One of my characters is suffering from this bug. My two other characters have received several Corsair Badges (as normal), but one hasn’t had single one dropped. So, there is a legitimate bug (as Buggie has listed on their issues list).