I ended up using a steel plate that I have for my shop press. I'm short on spare raw materials, so that had to do. I drilled a 3/32 hole, countersunk it, then slipped a finish nail through the hole. I sat the plate on top of my bench vice and locked the nail into the vice jaws underneath the plate. Then went to hammering on the nail head to form that rivet. It worked very well.
I drilled a 3/32 hole through the cap and stock, removed the barrel, finished the hole through the wood, then countersunk the hole in the wood and the nose cap.
I test fitted the rivet to get it flush inside the barrel channel, then marked the spot I wanted to cut the excess off outside the cap. I removed the rivet, cut the excess off, then annealed the stem that I'd be peening into the cap countersink, then reassembled it.
I locked the barrel in tight with a clamp, then peened the nail stem down into the cap countersink. Voila! I filed it flush. It looks pretty good.
Then, disaster struck. I removed one clamp that was holding my gun on my bench mount, went to adjust my headlamp, and the remaining clamp that was on my gun apparently had a bit of rotational tension and it tipped my gun over, rolling it right off my bench onto the floor.
It bent the !$@! out of my nosecap corner, knocked it out of whack a bit, and put a huge dent in the butt stock on the toe line corner.
So sad. I'm lucky it didn't break the toe or crack the forestock. I was able to bend the nosecap back a bit and I think I'll be able to file the humps etc out. But, it's going to remain out of whack and with a small gap under the barrel. That's a bummer.
Now... How to remove that huge butt stock dent?
The pictures are after I bent the cap back. It was much worse looking.