If the inside of the plate is flat, consider using a new (flat) carborundum knife sharpener (about 1" x 1/4" x 4") to plane off specific areas of the backside. Your idea of using fine wet/dry paper is also good if glued to planed flat sticks of wood (buy wood trim of desired shape/size at a hardware store). You seem to know where the inside of the sideplate is "tagging bottom", so taper your cuts towards those contact areas. After seating the sideplate, make sure that it is getting to bottom by gently tapping it down with a piece of wood. Also during each trial fitting, be sure to install and tighten all of the sideplate screws. You can use the white paint trick on these lower seating surfaces of the frame to see what is touching the sideplate (look for white spots on plate bottom).