"There is plenty of evidence including how the LXX of the same period treated the divine name"
The LXX and Christian writings are two different things. Judaism and Christianity are two different religions. Their books don't belong together. They were forced together by Christian scholars trying to ride the coattails of a well established and respected religion, Judaism.
The apostle Paul rejected them, or rather dishonestly re-interpreted the Hebrew writings for his own twisted purposes. But the Hebrew writings do not belong to Christians. So who cares what the LXX has to say about God's name. Christian didn't care about it, as it is shown by its absence in their writings