The manipulation of the Rune of Death or weapons formed out of it should also allow one to bypass Immortality Type 5 for being able to kill Demi-Gods who essentially lived outside of the concepts of life and death.
I don't think there is enough evidence to give type 5 immortality to Demi-Gods,
We know people can die in the Lands Between, and that explains why Those Who Live in Death exists.
Those Who Live in Death are blasphemers and goes against the ideals of the Golden Order as their souls do not join the Erdtree to resurrect or reincarnate, this contradicts the Law of Causation and Regression that the fundamentalists of the Golden Order love so much.
People affected by the Rune of Death die completely, their bodies and souls are killed, which again prevents them from joining the Erdtree and is against the ideals of the Golden Order, and the rune of Fia confirms that this order exists since Marika removed the Rune of Death from the Elden Ring.
All of these things also explain why the Dung Eater was given the death penalty, as he prevents the souls of his victims from joining the Erdtree, an unforgivable crime.
Enia also implied that Demi-Gods can die:
Well, well...I see...A remembrance of gold has found its way into your possession. Demigods, and even the greater of the champions, are hewn by the Erdtree upon their end, into remembrances. They are...valuable indeed. These remembrances yet house the power of their former masters. And should you wish to wield that same power, well... I will lend you the strength of the Fingers. Heh... Do not recoil from my offer. The Fingers guide us all. And you Tarnished, you are here to take, are you not?
But given their powers and their great longevity, none of the Demi-Gods could be killed or die, before the assassination of Godwyn.
If Demigods have Immo 5 while other people in Lands Between can die, that would mean removing or adding a Rune in the Elden Ring only affects Demigods, which contradicts other things established in the lore