Doug, you will need to have a cast flywheel from an engine using the same or similar ignition so that the magnets are the right strengths, sizes, and spaces so you will have spark. You will need to take it and your old one over to a machine shop and make them cut you the new keyway exactly the same going by the magnet's position. Then make sure they rebalance the flywheel to prevent vibrations.
I have seen this done before for some racing guys (they change cast to aluminum) or some pullers, too.
Personally I would leave a push mower engine running a pushmower. They are disposable engines.
If you are looking for a cheap small HP vertical crank engine, most areas have the old Snapper RER's running around or any old ride on lawn tractor that you can buy for peanuts and save yourself the pain of converting the push mower engine. They will last 10 times longer, too.
I'm not trying to be a wet blanket on your project, I'm just giving a different idea that might save you some time.