So I took a piece of half-inch thick aluminum scrap and used a hacksaw to cut a replacement handle. I then shaped it with a flat file and then cleaned up all of the rough edges with a belt sander.
(I know what you are thinking -- why not just buy a new plastic handle? I could, but I no longer trust those handles not to break at a crucial moment. I wanted a heavy duty handle.)
Next, I drilled a 15/64 inch hole in the center of the piece, starting with an eighth inch bit, then a 3/15, and finally a 15/64 bit. (Don't use a quarter inch bit -- the cutting threads on the tap will have nothing to cut if the hole is too big.) I then used a quarter-inch 20 pitch tap in order to create internal threads in that hole, to match the threads on the end of the rod.
I screwed the replacement handle onto the rod, holding back on the rod with a vice grips so I could get it very tight. The end of the threaded part of the rod projected out from the backside of the replacement handle by about 3/8 of an inch. I put a lock nut onto that little stub and tightened it, so that the new handle will not unscrew from the rod.
Here are two views of the finished product. This is a three-quarter end view:
And this is a side view:
