... the Order of the Bath and the Order of the British Empire each have a king of arms appointed who are not members of the College of Arms and must, therefore, be considered private officers of arms.
Indeed they do, so too the Order of St Michael & St George (though not the Victorian Order, which does however have a genealogist).
The present holders are:
Bath: Admiral the Lord Boyce; British Empire: Admiral Sir Peter Abbott; St Michael & St George: Sir Ewen Fergusson.
While these Kings of Arms are not part of the College of Arms (and indeed so far as I am aware have no particular background in heraldry - two were sailors and one a diplomat) and the appointments may be considered honorary, they can scarcely be described as Private Officers of Arms since the Sovereign of the Orders is HM The Queen. They would most probably be consulted by the College of Arms in any question that related to membership of their respective Order, but in matters of heraldic practice they would no doubt seek the advice of the College.