I'm in the middle of reading all the tablets at the moment, and I can't help this horrible feeling that the Labrys that can cut the strings of Fate will be used to kill Asterion in finality.
You could be right though, I believe one of the crimes Asterion was damned for was the killing of Tithonus, nd undoing Zeus' decree. However, Argos describes the act of torturing and killing Asterion repeatedly to be one of "divine worship", perhaps Asterion's first death was seen as a divine act by the Gods at that point? I believe in the line you highlighted, the host is Deadalus as the Labyrinth is his home, his creation, and so it is Asterion's bloodline which is cursed.