How can you sew a cauter pin into a belt? Answered

Acquaint yourself with someone in a machine shop.

Perhaps a trade school.

fairly simple task with the right tools and knowledge.

without the expensive tools, I would use a carriage bolt, a fender washer, a drill, and a hacksaw to make that hardware.

This stuff usually only has a slightly smaller hole in the leather and the backing plate on the metal bit about twice the hole diameter.
If you need it properly secured then simply add a piece of thin leather that you glue or sew in place to secure the backing plate.