A set of inexpensive macro focus rails will let you make fine left/right and forward/backward adjustments to the camera position. You'll still need to get the optical axis parallel with the plate, probably by loosing the mounting screw a bit and adjusting the camera. Once you've got it once, though, you can lock it down and use the rail adjustments to line the camera up laterally.
Also, using a longer lens will let you put the camera farther away. More distance means that you'll need to move the camera a greater linear distance for a given angular change, so adjustments don't have to be so precise. If you're using a 60mm macro lens, switching up to 100mm or 180mm will give you two or three times the distance, which should help make alignment easier. (And at 180mm and 1:1 magnification, you'll still only be ~18" from the subject.)