Were it me, I would just drill the rivet shank showing there in your picture with a 7/64 drill bit (if it were a 1/8th size rivet). The hole in the center is a good guide. Then crush down the thinned metal with pliers to narrow it on all sides. Next try to drive it through with a light hammer tap. May need a narrow, pointed tip type punch (like a leather punch) to finish the job. You could also make one out of an old round shaft screwdriver and a bench grinder. Just make the point long and tappered. (Does anyone remember ice "piks"?!)

But that is just me, I have a habit of finding the hard way to do things.