RE: Flat Plate
A flat plate will give you none of the requirements that you listed. The end grain balsa suggestion is by far your best bet.
A 12x12 single layer of cured carbon is 1oz, it would take 13 layers to make a 1/8" panel which is 13oz.
The a-sandwich is 2 layers of carbon, 2oz, plus your core 1~4oz, so your looking at 3~6oz (depends on the core material) with increased stiffness and durablility.