But the point wasn't just that of proving that Kirk was Kirk. It also involved proving that Lester wasn't Kirk - and that should
not be easy. Undermining the position of the ship's commanding officer should be full of obstacles instead, and Lester had studied Starfleet regulations with the intent of making things as difficult as possible.
As for talking a guard out of his duty... That really shouldn't happen. "You are to leave here at once" is right. But I fully agree that "You are mad" is not.
But the guards aren't complete idiots. They know not to take sides; they know not to pursue theories. They know that if they do nothing, things will get sorted out eventually. The CO can't e.g. execute anybody without their active cooperation, and despite appearances we never see the guards prepare for an execution. At most they walk people out of the room where a seeming madman has sentenced them to death - an eminently smart move!