I once applied 1 layer of carbon on the inside of an Antonio De Torres replica, you can do that with wetlay. That is a brush and epoxy.
If you want to make a 100% carbon rim (a.k.a sides), or even carbon with fiberglass, then you need a mold. The fabric that touches the mold surface will be the shiny, nice side so you need a negative mold. So it would look like the wooden mold you use to put normal bent wooden sides in (not my picture).
You seal it with clearcoat, then sand it 100% flat, then put release wax on it then you can use spray glue to attach fabric how you want it and then you wet it out with epoxy. If you want to do it without spray glue, you wet the surface out with epoxy, then put the carbon, and some more epoxy from the back. There are more elaborate and expensive ways like infusion but i would not recommend starting with that. A vacuumbag is recommended to suck the fabric against the mold.
If i was gonna do this i would build in a neck pocket into the composite component, to put a wooden neck in with its tail. You slide it in top to bottom. If you do this or not, you should get 1 continues piece of fabric that starts and ends at the neck pocket, so there is no visible seem.
The setup on this would be $25 for wood, $100 for carbon and epoxy, $50 in disposables (glue, gloves, mixing sticks, a scale, a container etc). I would recommend a fiberglass core between 2 carbon layers to save money.
Once you have a cured part, you open the 2 wooden pieces with the hinge, take it out. I would glue sandpaper on a board and sand both sides flush (with a facemask!).
You can`t rouyte binding channels in the carbon or your bit is dead. If you have a mold that flips open, you can build the ridges in the mold so there is a carbon edge to glue bindings in. It has to flip open for this or the part can never leave the mold. You can also just sand both sides flush and glue the linings in the body "to high". Then you can glue the soundboard on, and just route binding channels into the wood and not the carbon.