I don't believe Honda ever built a 12 volt GC electric start. As you have found, the electric start was 120 volt, for snow blowers. Not only will you need a flywheel with a starter ring, but also a charging coil and flywheel with charging magnets. As recently posted by Sonrisers, you can find flywheels and charging coils for GX motors on eBay, but Honda used at least 3 different tapers on the cranks. What James did was remove the magnets from the new flywheel, and drilled the old flywheel to accept these magnets. The GC and GS motors do have the mounting bosses for the coils, but they may need to be tapped.
Installing a GC motor is much more complicated than installing a GX motor, so be prepared to overcome many issues, of which there are no bolt on solutions.