The President should have sole authority with this position. Bgwah was a rather stubborn and unique case, there's no reason why the Senate needs to barge into a President's authority just because you lost a court cause to take over the executive decisions.
But if we are going to call this a meta-game position rather than an executive office, is there any particular reason why the President should have sole discretion over the Game Moderator as if it were another cabinet position?
As a meta-game position, it still must fit into the game somehow. This way we split the responsibilities between the President and Senate.
Yes, I know that it must fit into the game. The plan was to do so by treating it as a cabinet position. However, given how helpful the office, properly discharged, is to the function of the Senate, I do not see the special need of making it something that the Senate has no jurisdiction over.