P.S. I apologize for my constant upside down/sideways pictures, it only ever happens to me on here! 
Do you think the foam board would hold up if I screwed in some hooks to it to make it easier to mount to the door?Cut a backing for it out if pink foam board. Make the backing the proper shape to fit your door, then glue the broken arch to it and repaint. I would use construction adhesive to attach the pieces.