Legally, Musk has to pay Tesla for this, and someone very senior at Tesla (the board or the senior executive team) has to approve – without Musk being in the room. But, why wouldn't they say "yes"?
So I don't think this story is fundamentally implausible. (As I pointed out in a comment on another post, the "code review" was probably actually an "architecture review", but it is easy for non-technical people to elide the difference.)