So if it is common knowledge that killing someone tears the murderer/executioner/whatever's soul, then it seems unlikely that any country would utilize capital punishment because the cost to the executioner would be too high. Who would agree to be an executioner if they knew their own soul was going to be torn?
Back to the question at hand.
Sirius was convicted of murder not of attempting the Avada Kedavara curse but failing. It is possible that attempted murder, even by use of an Unforgivable, does not carry as stiff a punishment as actually killing someone - regardless of method.