Benjamin Thomson said that he found that adding strips of fiberglass-reinforced packing tape was one way to add stiffness. It may not be a permanent fix but you have plenty of chances for trial and error.
If the only pupose is to increase flexural stiffness the most effective place to add fiber is the furthest of neutral fiber = where board is thicker = on top of deck.
Check this thread out. You can route a thin slot on the deck and add carbon like this guy did (or regular fiberglass), then just add a patch over the strip of carbon. Maybe follow the stringers, but offset it.
I had a floppy longboard. I cut a trench 2 inches in from each rail on the deck then put some slow mix resin and some long strands of glass in the trench. Then a strip of cloth the length of the trench. Resin. Cured over a couple of warm days to make sure. Still has flex but no droop.
route a slot and glass in a stiff carbon rod (golf club shaft, old fishing pole) or a wooden dowel (if you want the board heavier). for extra credit pre-bend it (like a bow and arrow without the arrow) and glass it under tension, apex up. I did this on a fish once and it worked well.
or
add rail channels, but that will take some shaping then reglassing
or
always a good option…3) sell the board to someone lighter and make a new one. .
Just lam band of fiber on center of the deck, it’s the furthest point from neutral flexural axis. It will increase overall stiffness and buckling résistance of deck.
One of the material properties of carbon in a sandwich composite is it’s strength in tension. Putting it on the deck (compression side) may help somewhat but the bottom is where carbon tow or strips would do the most good.
I’d lay out a straight edge, route out 2 slots in the bottom and using epoxy, install some carbon tow. Lay some wax paper over it to smooth it down as it cures.
If it’s a light EPS core, it will still flex but that should stiffen it somewhat. Add more slots/carbon if needed. By putting the slots in the right place, you could fine tune your flex pattern to allow more or less flex in specific areas of the board.
I put a strip 300mm wide along the deck of a long board. So stiff it is almost flexless! Too stiff to be honest. Carbon is good in compression as well as tension. A layer on the bottom is good too. The rails are far too much trouble.
Lemat is right about the longest distance to neutral gives most stifness, so adding to the center is most effective. (basic material mechanics)
However, stiffening the center without stringer support increases the chances of buckling.
Therefore stiffening the rails would be a safer bet here if you’re not adding a fake stringer.
My approach would be routing a fake stringer in the deck. If still to flexy, I would add UD-glass tape on top of it (Carbon is not worth the price IMO). The fake stringer will support the UD-glass against buckling.