Engineering help needed

Bill and mjcouvi have the right idea.

Either a center board member to take the load. An L bracket int he middle, coming out at you to handle the load (or like 2 or 3 evenly dispersed). Or, remove the backing (likely close to card board / thin particle board) and actually nail in some real ply wood or something back there. I think that option might take a bit of effort.

Also, you could just replace the shelves with real wood boards and try to stain them or paint to match.

I think your OP idea is just a bit complicated and won't exactly work in the long run.