Eames LCW dating query  

I had a LCW, but earlier than yours, with five screws front and back on the underside, which was done the same way as yours. On my chair it was a repair done by someone who did not have access to a replacement oval shockmount, so they used two readily available round shockmounts.

What was done was that the original oval shockmount was cut away, leaving only a thin rubber layer left inside the inset cut into the wood of the backrest. Then, the repairman took a round forstner bit holesaw the same size as the round shockmounts he was using, and cut out round sections of the remaining oval shockmount with that. Hence, the thin rubber piece left between the two round shockmounts was the remainder of the original oval mount.

I know mine was a repair because the repairman did kind of a crummy job using the holesaw. Once the round shockmounts were removed, you could see the holesaw alignment and cut depth was sloppy. I should add that I suspect that the forstner bit holesaw was used as the primary tool to remove the bulk of the old oval shockmount. The repairman probably drilled out the ends of the old mount, leaving the part in between, then used a sharp chisel to slice away most of that part, leaving the thin piece between the two round replacement shockmounts.

I believe there were a few very early LCW that used two round shockmounts for the backrest, but the backrest did not have an oval inset cut into the wood on those chairs, as your chair does. I expect that for your chair a restoration would involve getting a replacement oval shockmount, and removing the old shockmounts down to the bare wood, and gluing the new shockmount in place. If your chair has fine thread screws, you might want to try to find an oval shockmount that uses fine thread, although I believe the coarse thread oval mounts are much more readily available.