Thanks, Colby. But... I did it! Here's what I did, with photo's:
-Practiced with a standard tiny rivet. It flared nicely.

-Needed the tail plate to lie flat, so I bored holes in a piece of pine board for the spool cap and click knob to recess into.

-Ready to start peening that click knob pin.

-Got it started with a pointed punch, then went to a 5/32" rounded one. But, I found that the wood was too soft, my peening driving the click knob into the wood, and the peened end through the hole in the clicker.

-So, I used a piece of sheet metal (a narrow scraper) and supported the tail plate with a piece of wood lath.

-The project had grown!

-The nipple on the round-end punch was too long and began to drive the peened end down through the clicker again. So I flattened the end of a common nail, and peened the end wide enough to hold the clicker in place.

-And it works -rather loudly. I also replaced the main gear and handle nut, as the threads for the handle nut were stripped when I bought the reel, with the handle held on with a large ugly nut. And another Streamlite received a handle from Dick, as well.

-I now have three well-functioning Streamlites.
Thanks to Colby Sorrells, John Elder, and Dick Janak for the parts, and the others who chimed in with advice and encouragement.