AnnOMaly:
If Josephus wrongly assumed that 1 Esdras 2's 'Artaxerxes' was Cambyses, how can you use his mistake as support for your position that Ezra 4's 'Artaxerxes' = Bardiya, or indeed that Ezra's 'Ahasuerus' and 'Artaxerxes' have to refer to kings between Cyrus and Darius? It doesn't follow.
I do not use Josephus as the basis for my conclusion. It just happens to say something similar.
How can it be neither? You are arguing for the first scenario, believing that Cambyses and/or Bardiya were also known as Artaxerxes or Ahasuerus (circularly arguing on nothing more than speculation or assumption). You can't point your finger at my reasoning, claiming it is circular because of favoring the 'out of sequence' solution, when your own reasoning is patently globoid.
I already explained that. The author of Ezra/1 Esdras (whichever was first) could have used the wrong throne name (which doesn't identify a specific king) but known which individual and period they referred to. This does not require that the king was broadly known as the other name. The author of E[z/sdr]a[/s] could simply have confused the names, just like Josephus does. Or he could have jumbled events from kings from both periods. Or he could have made the whole thing up about the letters. (How did the author of E[z/sdr]a[/s] get the letters? Are we going with the 'inspired' [aka 'magic', aka superstitious nonsense] theory?)