Prophet ‎﴾ﷺ﴿ Forgiving Murder
I think Prophet might have forgiven the murderer ONLY if Allah wanted to continue his test in this world. Prophet didn’t do things on his own, but based on guidance from Allah. I am sure this must be one of those cases where Allah may have asked the Prophet to forgive the murderer if he repented and accepted Islam .
I see it this way . Allah is the giver and taker of life, and Allah decides when the test of a person in this world is over. A murderer would, in normal circumstances be given death punishment. However, when Allah decides that such a murderer has repented for his crime and he has now come to accept Islam (after completion of proof), the Prophet may have been given the instruction from Allah that this person, who has now become a Muslim, and also repented for his crime earlier, deserves forgiveness. Thus, Prophet might have forgiven such a person. Allah forgives whoever he wills, except shirk.