When there is no clear heir to the throne, who is it that decides who gets to be the next king?

The only order of succession I really know of that's not by a close relative is a bit older than you're looking for. It's King Tut and those who succeeded him. Tut succeeded Smenkhkare, who may or may not have been his brother, but he apparently had no close relatives to succeed him and was thusly succeeded by his Grand Vizier, Ay. Ay was then succeeded by Tut's General of the Armies Horemheb. Now this is rather interesting cause Ay appears to have named Nakhtmin as his successor, but Nakhtmin never became Pharaoh. However, Horemheb appointed his own Vizier, Paramesse as his successor and he took the throne as Ramesses I (the grandfather of the better known Ramesses II, aka Ramesses the Great).