I think that @whrrgarbl has given you all of the correct steps for accomplishing this change. I just wanted to add a couple of additional comments regarding that answer:
I think your best bet is bullet-point number 3. Use wire cutters or a jewelers saw to clip/saw the ends of the stick pin as close to the pin's rectangular base as you can. Then use a jewelers needle file, or Dremel tool to file those ends so that they are flush with the rectangular base. Then you can use epoxy or jeweler's glue to affix a new butterfly clasp to the rectangular piece itself and you don't have to worry about un-soldering it at all.
I think you will be much more successful if you can afford to spend a small amount of money to purchase a few new items: a new stick pin1 , and, if you don't have a Dremel tool, an inexpensive set of jeweler's needle files. I just found a set on Amazon that are less than $10-I use a similar set that I purchased at a discount hardware store for about the same amount. They will have a finer grit and smaller points that are much more effective than any files you might already have in your tool kit.
1 In order to reuse the stick pin you already have, you have to remove it--with its base--from the pin it is currently on and then manage to get the base perfectly flat for re-adhering. Not an easy process.