No need for euthanasia,most scorps will take a freshly dead cricket placed in front of them if they are truly hungry(I partially separate the crickets head from it's body with my thumbnail,it twitches),they can go long periods without food,the biggest danger is de-hydration,without food they are dependant on the humidity and the moisture content of the substrate. If your scorp is sub-adult,it will regrow the missing limb in stages as it moults,full recovery will depend on which instar it was when the injury occurred.
The danger with leaving remnants of the old shell is that "fungus" may start on the old cuticle and infect the new cuticle,particularly in high humidity.
So best case scenario,you keep the scorp at the desired humidity for it's species and the old shell dries out and the scorp gets it off,worse case you leave it and the humidity means fungus grows on the old "skin" and your scorp develops mycosis (fungal infection,leading to poisoning). With a UV light mycosis will show as a black spot amongst the "glow".
It's not an easy question to answer without seeing it in the flesh. Myself,I would keep the scorp as is right for it's species,and check it with a UV torch,if there is a growing black patch on it's pincer,and you fear fungus is setting in,then amputation may be warranted,no need for euthanasia. If you were to amputate,make sure you use a sharp knife and cut cleanly at a joint