I don't know very much about the King James Bible. I see a lot of people who all seem to regard themselves as authorities contradicting each other's statements about how the King James Version was made, and as yet I have very little idea which of these people should be trusted as authorities. So please accept the following as the completely unqualified guesses of an amateur.
I don't see why Shakespeare couldn't have taken English Bible passages from the translators and improved them, made them sound more like Shakespeare. And many passages from the KJV do indeed sound a lot like Shakespeare. But I'm almost entirely certain that Shakespeare did none of the actual translating. Shakespeare seems to have had very little knowledge of non-English languages: a smattering of French and a few phrases' worth of Latin. Then as now, it was very unusual for an Englishman with almost no knowledge of Latin to have any proficiency in Greek, and even more unusual for a non-Jewish Englishman with almost no proficiency in Latin to know any Hebrew at all.
That's all I got.