Hot-setting rivets is fine at full-size because that’s a sizeable volume of steel to retain its heat in a process designed for speed.
I doubt it can be applied to the tiddly things we use. The rivet would cool to black before we can insert them and bring the snaps anywhere them. Heating them in situ will of course heat the surrounding steel, risking all sorts of harm.
If the rivets are for show only and the reverse will be invisible, use those round-headed slotless screws in tapped holes, with locking / sealing compound. (I think they have a tiny hexagon for a spanner.)
If they are close enough to scale, consider those button-headed Allen screws. They are made only in metric but M5 is near enough to 3/16″ diameter here. Painting will fill the sockets – the larger ones might want filler as well. I used M5 ones for the rear ring of my steam-wagon’s smokebox, tapped into the ring. You can see only the upper ones, and once painted their real identity is sufficiently camouflaged. The front, or door, ring is held with 3/16″ rivets lightly hammered shut on the inside – they take no significant loads, and those in shear. The heads of these different fasteners match closely enough for aesthetics, given that the rings are several inches apart and the bunkers obscure most of the smokebox.
Now, that’s for hidden reverse applications. What of visible reverse sides?
We can buy only a limited range of those slotless screws. I wonder if hemispherical nuts to match would be feasible, short of putting standard ones on a little mandrel in the lathe and form-tooling the radius. Or making them from scratch. (Use a ‘Dremel’ or similar with small cutting-disc to trim the protruding screw.)