I have used, well a lot of kinds of rod for rivets in my life. Nails, various bars, steel, copper and brass one ended rivets.
But,............ for a first timer..........? cutlers rivets are not that expensive. Try Lee Valley (.com) These are 2 part, swat them together one time and they firm right up and lock.
Just take off the handle and use it for a pattern to rough trace out your new handle.
Just trace it oversized for now.
DRILL the rivet holes first!! Mark though the frame and spot your holes.
Trying to place 3 holes perfectly -after- you cut it out, is............. well, forget it.
Your best shot is drilling the holes first and then fudging the pattern to fit.
Drill your new wood softly, especially when you are about to break though. The last wood your drill bit finally breaks though, is the side that shows in the end. So easy does it.
I like to assemble with 5 minute epoxy. From the hobby shop. It sets fast, but best of all, its a slightly flexible resin. That means it grips and absorbs shock better.
The slower the set time, the harder the resin. But you don't want hard. You want grippy and flexible.
5 minute epox is good stuff.
rasp, file, scrape, sand and polish as you like.
yours Scott