How can I convert a stick pin badge to a butterfly clasp?
For removing the stick pin backing, you need to remove the parts that stick out and file it down to be smooth. To remove, I'd use one of the following:


a Dremel tool with appropriate bit
if it looks like the badge was soldered, a soldering iron to un-solder it. (Make sure that the front of the badge won't be damaged by the heat, and follow safety precautions! If you guessed wrong and it was epoxied, heating it may release toxic fumes.)
a pair of wire cutters, depending on how substantial the pin back is. (This would work well on the butterfly clasp pin pictured, not so sure about the stick pin version.)


Then to smooth the back, you could use a metal file or a Dremel tool with a grinding bit.

To attach the new pin, I would just buy a new butterfly clasp base at a craft store (example). If you really wanted to re-use the clasp from the other badge, definitely use the Dremel tool, as you will need to cut a base for the pin and not just clip the straight pin off.

Then you just need to epoxy or solder it to the now-smooth badge back!