Assuming Charles lives long enough to reign, he has more than two choices as to has regnal name. His name is Charles Philip Arthur George. There have been many that have said over the years that he would choose George VII, after his grandfather. Many people think Charles is a cursed name. Whether or not you believe in curses, it's possible that there's a belief that Charles III could be the last monarch of Britain or the Commonwealth. King Arthur isn't reasonable for obvious reasons. There is also the possibility of King Philip, in honour of his father. It's ultimately his choice when he takes the throne and he's not necessarily limited to his birth names, although that has been the custom.
When you think about it, it had to have been jarring in a way for so many when Princess Elizabeth became Queen Elizabeth II. Before her, there was just Queen Elizabeth, not Queen Elizabeth I.
I have had a thought for a long time about the monarchy. If it continues to survive, it will likely be a very long time before there's another Queen. Unless there's another tragedy, it will likely be beyond the lifetime of anyone here.
Of course some in Scotland insist that she *is* Elizabeth I (though that's a whole new can of worms)