Hi Scott,
"If the King is 'a star' and the shafts provide a 'window of opportunity' to rise to the star, then which star is Khufu rising to since both Mintaka and Al Nitak pass over the line of sight within Khufu's potential reign?"
In the Dyn 5 pyramid of Unis, the king, purified himself with the cool water of the stars and bathed in the starry firmament §138, the 'Imperishable Stars' raised Unas aloft §139, and Unas guided the 'Imperishable Stars' §373. Unas was born as a star, and he appeared as a star §262. Unas opened his place in the sky among the stars of the sky, for he was the Lone or Unique Star §251. Atum, the 'father' of Unas, assigned Unas to the excellent and wise gods, the 'Imperishable Stars' §380. He was a son of Sopdet (personification of Sirius, the brightest star in their sky) §458, and was given a warrant as Great Power by Sah (Orion), 'father of the gods' §408.
In the PTs, there seems to be at least two threads that associate the king with Sah (sAH, 'Orion'). The dead king was identified with Osiris who in turn had an identity of the one in Sah, and the king was also born as a star, appeared as a star, and traveled the sky with Sah, as a companion of Sah:
"In your identity of the one in Orion, with your season at the sky and season at the earth: Osiris, turn your face and look at this Unis, your seed that came forth from you active: he will live and this Unis will live, he will [not] die and this Unis will not die..... W 152 (PT 219 §186a U; I:106)
"The sky shall conceive you with Orion, the morning-star shall give you birth with Orion. Live! Live, as the gods have commanded you live. With Orion in the eastern arm of the sky shall you go up, with Orion in the western arm of the sky shall you go down. Sothis, whose places are clean, is the third of you two: she is the one who will lead you two in the Marsh of Reeds to the perfect paths of the sky." P 38 also. N 9 (PT 442 §819)
"Ho, Pepi! You are the big star that is Orion's companion, who travels the sky with Orion and rows the Duat with Osiris. You shall emerge in the eastern side of the sky, renewed at your proper season and rejuvenated in your time, Nut having given you birth with Orion, the year having put your headband on you with Osiris. P 317 (PT 466 §882)"
I think it might be better to think in terms of regions of the sky the AE associated with Orion at its highest point in the south and the circumpolar stars in the north - these regions being 'enclosures' or 'mansions' of the celestial gods and also in the northern sky, the 'Marsh / Field of Rest / Offerings' - the region of the circumpolar stars. Thuban and the 3-star asterism in the centre of Orion may have simply designated these 'regions' that the king could 'visit'.
CT