[ossiblue]

My guess is the frame is a dense foam-type material rather than wood and hanging with wire puts all the stress on a single anchor point--the screw-eye--and the stress is angular as the wire pulls toward the center line.

Looking at the picture, is there a cross member to the frame at the bottom of the arch? If so, that's where I would secure either the hangers in your first or second picture--across the horizontal member. All stress will be perpendicular to the anchor (nail or screw) with less likelihood of pulling out. If there is no cross member, then use the hangers in either the second or third picture, and mount them separately to the wall--could be a PITA to get level. You could also use a flat stock piece of aluminum running horizontally, secure it to each side, and mount the aluminum bar, level to the wall.