I decided to repair a rusted-out mounting bracket on the left-hand running board by transplanting a bracket (and the surrounding sheet metal) from the other board, which is too messed up to be worth restoring. I cut away the bad part, and trimmed the donor piece to fit perfectly in its place. Then I welded it in place. The welding caused the board to warp a bit, which I will have to fix by un-shrinking the weld beads.
To replace the unusable right-hand board, I am using an old repro board that is missing its mounting brackets. I made a new pair of brackets from some U-shaped metal channel, and drilled and filed the keyhole-shaped holes for the bolts that attach them to the car’s frame. In the next few days I will weld the new brackets to the board, and add some additional bracing to compensate for the fact that the repro board was made from thinner sheet metal than the originals. Then I will be able to sandblast and paint both boards, mount the rubber sheeting to their top surfaces, and put them on the car.