You could just buy a sheet of carbon, cut it to size then bond it to the wood. This will be more expensive than doing it yourself and it might add more thickness than you want, but it will be quicker.
If you want to have a go at doing it yourself, I would cut the wood then lay the carbon down, if you route it after bonding the carbon, you risk delamination.
Cut your carbon with the same template, but make it slightly bigger, maybe 20-30mm bigger. You might want to base coat the plywood black (not the edges), otherwise you might see the colour of the wood show through the fibres. A better option is to give it a very light coat of epoxy that has been tinted black. The purpose it for it to bite into the wood, not to wet the carbon so wait for it to go tack free, then lay down your cloth very carefully so it doesnt distort, you could apply some spray adhesive or hairspray to the back of the cloth so it stays together better. Make sure there is no trapped air. At this point you could use a roller and lightly go over the cloth so it holds the epoxy under it and doesn’t move when you apply the clear. Now go over it with some clear epoxy. Light coats is better so you don’t build lumps. Try to wet out from the middle out. Using a squeegee like you suggested is fine, but it could distort if you are heavy handed. A roller or even a brush is fine too, whatever you’re comfortable with. When you get to the edge, try to wet out about 10mm from the edge, leaving 10mm dry, but make sure the edge is wet. The idea is to give the bit hanging out a bit of weight so no air can get in, but not too much weight, as this will create a pocket just behind the edge. Make sure its all wet and be careful not to distort the cloth. Keep an eye on it, and wait for it to gel. Now you can easily cut the excess cloth with sharp knife or even a pair of scissors. If you wait too long, it’ll be rock hard, and youll have to use a die grinder to cut it. You dont have to worry too much about distortion, as the resin has set and is holding it in place. Leave about 1-2mm. Once its cured, you can sand down the 1-2mm you left with some 120 then 240. Now you have 2 options. You can hit it with another layer of epoxy if you want more depth, or you can clear coat it to give it UV protection. Or you can get the best of both and give it a coat or 2 of Durepox. Its an epoxy urethane that has good adhesion to carbon, it builds, has UV resistance, and it will seal the plywood edges. Dont go mad as it has a yellow tinge, but even at 2mm, you cant notice it.
I think it will look quite cool!
