I was just comparing the inside of you FA to an AE-1, the mirror on your camera has clipped corners, plus that little square tube bar that protrudes through past the mirror. Wonder what those are about, and if they are the reason your lens won't function. Maybe try a zoom or short tele to allow the mirror to swing.
Doh, just realized the mirror on your body is longer than standard, comes down farther in front of the aperture control tab.. Maybe shorten it or replace it with one from a donor body.