I don't quite understand your arguement for exp of a complex line to be totally geodesic. But if so there is a lemma in vol II of Kobayashi Nomizu (Prop. 7.1) that says if two tensors with the symmetry and J invariance of a Kahler curvature tensor agree on complex lines they must be equal. It's clear curves have to be flat by Gauss lemma and totally geodesic would mean the second fundamental vanishes, so we can measure R of the ambient space from such curves. Thus the Prop. says total curvature must vanish.