Well, the bit gets set during both instances... that's the reason why I mentioned it. They don't get unset in either circumstance....

Fixing the Bow+Arrow circumstance might be complex. Bonuses and properties are set when both bow+arrow are equipped. If either component is removed, these bonuses/properties are removed (except for the long range bit and maybe others, like the Lightning Arrow).
For the ambidextrous situation, it is arguably far more complex... I think it's supposed to be handled like the bow+arrow situation, except you just need to add back the properties of the lone weapon on either hand (bow or arrow solo = 1 attack power).