Hi, Mel! Right now, Wonder Woman and Superman are discussing using magic to clear the public's memory, but as you can see from my post, they won't take that route. My idea for how the heroes wind up wiping the incident from the public memory comes from the 1978 Superman movie. After Batman and Cyborg trace the location of the broadcast, Supergirl flies around the earth to turn back time to slightly before the broadcast went live. With the knowledge of where Harley Quinn and Poison Ivy are hiding, she can get to them before the broadcast, and deal with them then. To solve the issue of them just shrinking down again to break out of Arkham, we can maybe say Kara confiscates the last of Poison Ivy's shrinking toxin.
Here's the clip from the movie, if it helps:
https://www.youtube.com/watch?v=LoCaeI5RffI