I think you could likely solder a new piece on, but you'd need to disassemble the knife enough to get that safety lever out
DIY
For DIY - this is also a placeholder.
The knife can't be disassembled, it's riveted together
You can absolutely disassemble riveted parts, you just need to have new rivets, or try to salvage the originals. This is obviously not an easy process, but it's probably the best chance at actually soldering a button on successfully. Another option to consider is if you prepped the two tiny tabs of metal with solder and flux and tried to solder a new button on by heating the top. You would need a screaming hot iron to heat it enough, and you'd only have a short window to solder the button before the handle scale underneath burns. Otherwise you can try and glue something on, but you'll have a hard time not gluing the whole thing closed and it might not be strong enough to handle floating around in a pocket