Yeah Peacock would lose but I think she would keep fighting him to steal it for herself. That would be his story, the mask winds up in New Meridian and everybody fights for it before it gets stolen or tricked into taking it off. As for Batman his story would take place after Injustice and instead of going back Earth 1 he wound up in New Meridian. Upon hearing about the skullheart he goes after it to wish himself back home. How? Batman!