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It would be King Arthur (becoming Athur I if another Arthur were to become king after - they don't give a number to the first until there's been a second, hence Elizabeth I or Henry I, but just King John and Queen Anne). It would make no committment either way to Arthur Pendragon having been real, because the numbers only count for since the Norman Conquest in 1066 (for the same reason if they call it Edward, it would be Edward IX not Edward XII, because of the previous 11 kings called Edward, only 8 were post-Conquest).
If Charles and Elizabeth are out, wouldn't Philip have the same problem?
As well as King John, Prince John (George V's youngest) added to the perceived "royal curse" element of the name too - can't see them using that at all.
As a Brit I wouldn't find Princess Lexie-Mae too contentious at all - Lexie is from Alexandra, a common royal name, and Mae is only an alternative spelling of May, again a common royal (nick)name (it's how Mary of Teck was known). Lexie-Mae/Alexandra Mary would be no stranger than Harry/Henry, Kate/Catherine.
Prince Kenzie-Lee on the other hand would indeed be a bit more ~out there~.