First, leave the flaps as a sheet, don't taper it towards the trailing edge.
Apply a coat of dope top and bottom to seal the wood.
Cover it top and bottom with the lightest carbon fiber mat you can find and squeegee most of it off, leaving just enough to make the surface slightly wet. Use the thinnest epoxy you can find(West Systems 206 is pretty good- 2 hr working time) or warm the epoxy with a heat gun to make it wet the CF more evenly. Use at least a 30min working time and do each flap one at a time.
Clamp each flap between two very smooth, straight, flat planks(almost anything including a granite counter top, MDF is cheaper) with 6 mill or so polyethylene sheet top and bottom. Make sure it lays reasonably flat. Don't want to make a warped flap. We don't want to glue the flap to the planks.
Keeping the flap the same thickness all over essentially makes it into a box beam, which is stiffer to twisting than a triangle shaped beam.
The carbon fiber, even the <.4 oz/sq yd stuff, is much stronger and stiffer than anything else.
You can also use polyethylene to spread a fixed volume of epoxy over a fixed area to make all four sides similar in weight and thickness.
You can get pretty good results with dope instead of epoxy. It will be a bit lighter. The dope won't be as stiff as epoxy, but the carbon fiber is MUCH stiffer than silkspan.