This is a question that was not answered in the show because it wasn't the point of that episode. The point was that Kurn, who had helped bring Gowron to power, was sick of being dishonored because of his brother's acceptance of discommendation in TNG "Sins of the Father". In that episode, it is revealed upon examining data having come from a Romulan vessel, that Mogh did not send out the coded message. It was, in fact, Ja'rod, father of Duras. Mogh was used as a scapegoat, as Duras' family was too powerful and it would cause a civil war within the Empire. The thought behind Kurn having his mind erased and given a new family was that he could reclaim his honor in another way, and to show that sometimes there is no "good" or "easy" way to go about things, and that those actions also have consequences, but, the reason as to how and why this can be done is a matter of a lot of debate, I would Imagine.
There is not a lot on exactly HOW the honor system works, other than what we've seen in the shows so far, but we can come up with a few conclusions based on what we do know. In the DS9 episode "Homefront" we are given some info concerning Klingon Gods.
"Our gods are dead. Ancient Klingon warriors slew them a millennia ago. They were... more trouble than they were worth."
In most religions, one's soul (although their own) typically belongs to that of whoever/whatever is believed to be the God/Gods of the religion. Since the Klingons killed their gods, it can be concluded that Stovokor and Gre'thor are places like Heaven and Hell, places which cannot be proven by hard evidence as to exist or not. Since the gods are dead, these places may still be part of the Klingon psyche and used to further iterate the importance of honor. It was insinuated in the VOY episode "Barge of the Dead" that B'Elanna did go to Gre'thor, but even then, her mother was saved not by using rituals, but by B'Elanna being true to herself and living a good life, which would further denote that the honor system is subjective and based more on personal and societal norms for their culture, which may also mean that by Kurn being "made" into this different person, that he no longer carries the burden of dishonor that Worf has to, as he no longer has the knowledge of who Worf or Mogh are and, from his perspective, is not part of their family. Not having to carry the burden any longer means that Kurn would not consider himself dishonored, and since the term is subjective, he would have no reason to suspect that he would end up in Gre'thor, and therefore would not go there upon his death.
In closing, it's been established that there are no Klingon gods anymore, and as such, conceivable that whatever power the gods had over their souls now belongs to them. The honor system and the way in which they treat honor also does reveal that depending on the circumstances, it is a subjective and fluidic concept, and more dependent on the person and society than anything else. Kurn's soul therefore would be intact because he has no knowledge and does not carry the burden he once did. He had been given a new name and a new family, one who's reputation was not tarnished, so per the lore established, it can be concluded that his soul would be clean. In TNG "Sins of the Father", it is established that the dishonor would carry down 7 generations, but this would most likely be more due to the fact that the family would be made to remember, as it would be noted by the High Counsel and the burden on the Family's souls would be made to be carried by the family members.
Side note: Although in terms of lore, this answer works, it's also probably just due to the writing staff thinking more along the lines of "well if they do this, Kurn won't have to live with the dishonor" and had not even thought of how it would play into going to Stovokor or Gre'thor.