There is a type of translation that just something else translated. To whatever extent King James did that with the Bible, he is not responsible for it. There is also a lot of judgement that has to go into translation, and also King James didn't really translate everything so literaly, so in some sense you can say the King James Bible is inspired by the "real" Bible. In which case King James is responsible for his version of the Bible.
So in that manner I would say that BB owns his sonnets if they were merely inspired by his muse. If his muse was actually dictating and he was just translating thats a different matter.