There is a real trick to peening to make rivets or what ever. I learned that is particularly so with very thin stuff, like 1/16" RR pins. The trick is to cut the long end off fairly close to the countersink.
Keep it as close as you can so the end will mushroom, and won't have the tendency to bend over. With thicker rivets like MC rivets, I used bare #8 copper wire. Copper because it provides a nice color contrast that looks cool, and, it's very soft and peens easily.
Cut off to be close to your finish plane level again, but in addition to the MC metal side, put a slight chamfer in the wood on the inside where the barrel will be an anvil for your hammering. That way you'll get some mushrooming there too to give you a bit of a mechanical lock. If it starts bending on you STOP and get it out of there. Start over with a new piece. It's almost impossible to stop it from bending over on you by directional hammering or trying to bend it back and it'll just wreck the metal side. Trust me. I learned this the HARD way.
On the peening / hammering side, (with fat stock like # 8 wire) I start in the middle of the pin with a metal centering punch or awl. The wedged nature of it helps drive it down in that thick wire and get the mushroom started. It also gives you a measure of control to make sure the mushrooming is somewhat symmetrical. It doesn't really matter how thin the shank is relative to the head size. Just a little bit wider with the head is enough to provide the mechanical lock you're looking for.
Once it appears to fully fill the chamfer give it a few more taps and then stop. If you bang away further you will just widen the chamfer and make a dent in the surface you are trying to put the rivet head over. The operative word is to STOP when it's right. You're going to file the hammered blob off to being smooth and flush anyway.
File until flush and then polish with progressively finer sandpaper.
When done, drip on a couple of drops of thin ACC cement to help with the mechanical lock.
Stand back and admire your work.