Now, I am weak on religion. But, isn't there an idea that whatever the Pope holds true on earth than God will hold true in heaven. So, maybe the King-figure has to be blessed by a religious figure for him to be a true king by heaven. And these are the kings whose descendants can be KotC.
That is a common misconception about the nature of the Pope's Infallibility (which was taken to extreme levels of ridiculousness in "Dogma"). Boiling it down as much as possible, the Pope's declarations are only held infallible as they pertain to the Church itself, and/or are made ex cathedra (when he's sitting on the Holy See, his pretty awesome throne). Still, I'm quibbling, and I think your point may be valid.
Saladin had the support of both Arabs and Muslims (not the same thing), and certainly would have been blessed. None of my primary sources (wrote one of my theses on Saladin) described a ceremony, but I'm sure one took place.
George Washington is the 14th great-grandson of King Edward I (the Longshanks; the bad guy from "Braveheart," which was fun but a total bastardization, as I'm sure you all know. Fun fact: one of my ancestors was knighted by Longshanks, and later was made a feudal baron, though a tiny and relatively insignificant one; technically he was a Baron, but not a Greater Baron (since he didn't inherit the title), and was referred to as Lord of the Manor.) His reign predated the Church of England split, so his rule was "blessed" by the Pope (and he wasn't the head of the church himself), though he never met him personally; it was a proto-divine right permission-type thing.
Shiro was a descendant of the Last King of Okinawa, right? That's Sho Tai, if memory serves. I think they were Confucian, and I don't know enough about their clergy to discuss it with any authority. There was a concept of Heavenly Mandate, as far as I remember, though I have no idea what the Meji Restoration did to that concept, and that's right around the reign of Sho Tai (which was cut short; that's why he was the Last).
Butters's ethnic background is important—if he's Ashkenazi/Khazar, ethnically descended from the Tribe Judah, or if he's descended from converts. If his background is from Judah, it's possible he's descended from David or Solomon.
Oh, and Charlemagne was personally crowned by the Pope, so that one's easy.