Not only do the ring gears have a different number of teeth but the flywheel diameter where the gear mounts is different so you can't swap ring gears on the same flywheel. You'll have to come up with a 12 Volt flywheel.
What you might be able to do is mount the 12 Volt solenoid on a 6V starter and run the starter on 12V. Running 12 volts through 6V starters is commonly done. Mine has lasted 25 years and 100,000 miles like that. You might have to come up with a solenoid-mount starter or drill holes in the starter to mount the solenoid. I don't think the stomp-starter switch holes will line up with the solenoid holes.
