Though I don't usually answer my own questions, I came up with a simple solution today while riveting a mailbox: Use a flat-head nail to pull the bulging side of the rivet flat.
- Before riveting, find a flat-head nail with shank of the same diameter, or just slightly smaller, than the "nail" that is part of the rivet. You can push the rivet off its own shaft to check that the nail slides through freely, then put the rivet's "nail" back into the rivet.
- Rivet as you would normally.
- Use the nail or a punch to pop out the part of the "nail" that breaks off, remaining in the rivet.
- Put the flat-head nail through from either side of the rivet, and use the riveting tool to pull on the nail as if it were the rivet's shaft. There's no need to pull until the head pops off, as long as the bulge flattens neatly. This is surprisingly easy to do, especially with soft aluminum rivets.
- Remove the flat-head nail.
- If need be, put a dab of glue or paint in the center to seal the hole.
Why not use the just the flat-head nail, rather than go through a two-step process? Those attempts left the rivet incompletely flattened; the original "nail" is designed to flare the hole, which I found is needed before flattening the head further. (Mileage may vary -- if you have a stronger nail and riveting tool you might be able to avoid the two-step.)
From now on, I'm keeping a few suitable nails along with my rivets.