Welcome,
Is there a reason you don't replace the engine with another GY6 type engine? The 420cc is only supposed to be run @ about 3600rpm, That's not a lot of wheel speed- it makes lots of torque. The only gear box I know of that is "affordable" is:
https://www.monsterscooterparts.com/...kaAtBEEALw_wcB
It says it's not been rated for larger than 6.5hp- I'm guessing no-one has reported back to the manufacturer about using it with larger engines. It's for the TAV2 30series CVT system.You'll need some type of CVT sytem to get power to the axle and have some type of speed.
IMO- I would try to find a CN250 or similar to swap out if you're looking for more power.
Keep us up to date on the swap out so others wanting to do a similar modification will have some feed back about the performance and all that's involved.
Not having your buggy -I can see lot's of fabricating with this swap (the 420cc), engine mount,axle mod's for the TAV2 system to match the sprocket location (if it's adjustable), different sprocket and space for everything to fit in the original swing arm, then the space for the reverse box,wiring, ect.....
If I knew that much fab. work was needed I'd be putting an ATV engine in it!!!But that's just me.