Starting from this claim that the king of Persia respected the Jewish' God.
The only surviving text is the Jewish text.
Are there independent sources of ancient Persia's record corroborating this?
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If you are expecting a source against Darius the Mede, as noted in the Bible, then you may be sorely disappointed since there are no primary sources that have made any connection between the king as noted in the story of Daniel and any living king of Persia. If you can't find a primary source on the king who issued the edict then you are going to have a hard time finding material for or against something like that proclaimed by any king of the time. Flavius Josephus is the only one who mentions Darius the Mede; Josephus' research is more focused on Jewish History of the time so it's not going to be an independent corroboration, and even his mention is unable to be linked to a known ruler.
The only record of the time you might be referring to I have seen is for Cyrus the Great, where he allowed the Jews to return home and practice their religion. Although his general attitude was to allow worship in countries he conquered and ruled, there is an overview of the Jews in the Achaemenid period at Iranica Online which gives an overview of their history in Babylon. If you read in between the Jewish sources you can see the other historical records there and the overview seems to be one of general neglect on Jewish communities so long as they went along with the government at the time.