I am not certain whether the OP’s goal is to create angles with high precision or to display existing angle values to greater precision (or both?).
When @john_mcclenahan and I developed Angular Dimension 2, we followed the units setting from the model info. We had, as I recall, three things in mind: consistency with how the user has told the rest of SketchUp to display angles (e.g. protractor); avoiding making our already complex settings dialog even messier; and concern about how adding more digits might wreck the placement of the text in certain cases (John McC tore out a lot of hair getting it to work reasonably well!).
But, as @john_drivenupthewall has shown, that’s all a matter of formatting the angle’s value for display; the internal representation is quite precise. Our extension’s Ruby code is not encrypted. If someone wants to tweak it for their own use, that’s fine with me.