The envelope theorem says $dV^*/d\theta = \partial\mathcal{L}/\partial\theta$: only the direct effect counts. Why can the indirect effect be ignored?

D. Because the first-order conditions already hold at an optimum

You are already at the top, so moving slightly sideways costs nothing to first order. This is why a shadow price can be read straight off the Lagrangian without re-solving the problem.