I agree with the added two inch rip on the back, just make it the same length as the shelf and keep the top edge flush with the top of the shelf and it would be not so noticeable from the front,or place it under the rear edge of the shelf kinda like a pinrail and attach with a few fine finish screws, also if possible don't make the shelf "too" much longer than the center so the load bearing would be closer to the sides thus avoiding the laws of gravity.
You could always use a rip of Birch, Oak and or Maple ply wood you would just have to wrap the edges with a pre-made trim stock to hide the edge grains showing the ply layers.