It will be hard to save a rivit that is fully set, the factory setting process swells the rivit to fill the hole---unless it is being used as an axle or pivot.
You need to say more about how the rivit is used, perhaps pictures.
I have many rivits, but not always what I need and have saved some, one of my projects is a 53 Citroen, I am working on some seat brackets that have rollers rivited in, the end is upset just enough to keep them from falling out, I used a dremel to remove the little bit that is mushroomed over --now I can paint the bracket grey, have the rivit black, and clean the stiffness out of the roller and lube it---a lot of work just for a seat bracket.
On another project that had a rivit with a special head, I removed the mushroomed part, drove the rivit out saving the special head, then welded a new piece of metal on the other end to re-rivit, also a lot of work, but I got it done.
I have also taken a bolt, made the head to look like the rivit, then used a nut.
Post a picture of what you want to fix, then you will get more ideas from others posted.
