You need a punch slightly smaller than the small end of the pin and a vise opened just enough to clear the pin head to support it as you drift it through. It's not a rivet. The pin is only press fit into the thin back wall of the case. Tap these 2 pins out and it all comes apart. The one here is a later type than yours with the plastic cover, so there's no face plate and the pins don't have the wide heads. So inside the case, you have the trigger, the cam plate, the pawl, the mousetrap spring and 2 pins. There's a 3rd pin holding the case together, but you don't have to bother with it. The cam goes on the trigger in this direction, pic 2.
It might be best to disassemble yours before soaking. Your face plate has the printed on logo instead of the earlier embossing and I'm not sure how it would react to the Evapo-Rust.
Getting it together again is the hard part because it's under spring pressure. After some wrestling, you will have all the parts lined up and both pins poked lightly in their holes. Then, lay it down on a hard flat surface and tap the pins home.
edit: forgot to mention. I find it best to locate the pin that holds the spring and pawl first.
trigger001.JPG
trigger002.JPG