If you were making a flywheel that is meant to store energy, you would want it to be made out of a nice heavy material to avoid having to spin it too fast. For this reason, carbon fiber is probably the WORST material that said flywheel could be made out of.
Maximum energy stored for the weight necessitates high rpm. Disintegration on overspeed is the problem. Steel becomes shrapnel. Carbon fibre becomes fuzz. For the weight carbon fibre should be able to store more energy than steel as it will be able to spin faster before destruction. The wheel is shaped such that a planned band near the rim is formed with less thickness so that on overspeed only the rim disintegrates. Like a shear pin.
Some work was done to in the late 70's to incorporate a flywheel in a coal mine in Pennsylvania, I believe. The shuttle car was to have a 5 flywheel package where the cable reel was. The flywheel would provide thru a gear box hydraulics, lights, tram and conveyor power for 20 minutes. Idea was at the feeder when dumping coal to contact the power supply and spin the wheels back up. I never did hear how the tests went. I saw that in an old Mechanix Illustrated. Since 30 years from that I've never heard of such in the coal fields I'd say that normal mining "stop and go" methods proved the idea not feasible in that application.
I think some trolley cars on the west coast were using flywheels to go beyond the trolley wire a short distance.
Still, the fibreglass and Kevlar wheels could hold considerably more energy per pound than steel because they could be spun faster.
Being in a car you would have to deal with gyroscopic effects. Friction. Shielding if the wheel fails. Power input and ouput. These things at speed are literally a bomb and must be respected as such.
I have sought some means to keep the speed down and yet still store energy. I have thought of a tube with a piston pressing on a gas that when sufficient pressure is achieved that a phase change would store energy toward the rim as it pulled a vacuum on the axle side. Or vice versa. Thus at a given temperature the wheel would move the piston back and forth thus altering the rpms of the wheel.
I thought of using magnets in a similar manner. Mount a permanent in the outer end of a pipe and another that when at speed is centrifugally pushed toward the end one and is repelled as the two North ends face each other. The faster the wheel the nearer the magnets get and the more repulsion they feel. Or somehow use electro magnets in a similar way.
The goal is to pump energy into the wheel other than kinetically. The main storage would be in a narrow range of rpm's. Yeah that be the goal.
Just some thoughts.