You might get some ideas of what to use by looking at other hardware screwed into the outside walls, such as window blinds/curtains, dinette table brackets, etc. On my motorhome, these are generally about inch long screws.
Don't expect a vast amount of holding power from the wall structure; it's probably just luan (cheap thin plywood) attached to a styrofoam core. A well nut or some other sort of expanding fastener would give greater holding power. (It sounds as though your stress will primarily be in shear across the screw, parallel to the wall surface, which doesn't require too much holding power.)
