That works. The shittiest supervillain ever has the shittiest origin story.
Trump champions the birther movement, which made no sense on any level but everyone eventually talked about it.
Obama tries to reason with it at first, eventually gives up, shakes his head, and shows his birth certificate so that everyone can move on with their lives. Obama cracks the odd joke about the whole thing but eventually moves on.
Obama spends a few minutes making fun of Trump at a dinner.
Trump drops everything and DEDICATES EVERY MOMENT OF THE REST OF HIS LIFE TO UNDOING EVERYTHING OBAMA RELATED EVER RAAHHHHGHGHGHGHHH.