= Solution
Let $|\alpha_1^\perp\rangle$ be a unit vector in $S$ orthogonal to $|\alpha_1\rangle$, and choose
$$
\Pi_0=|\alpha_1^\perp\rangle\langle\alpha_1^\perp|,
\qquad
\Pi_1=I-\Pi_0.
$$
Outcome zero is then impossible when Alice sent $|\alpha_1\rangle$, so it identifies $|\alpha_0\rangle$ without error. This measurement succeeds with probability
$$
p_1+p_0|\langle\alpha_1^\perp|\alpha_0\rangle|^2
=p_1+p_0\sin^2\theta
=1-p_0\cos^2\theta.
$$
The optimum under the extra requirement is at least the performance of this admissible measurement, so
$$
\boxed{P_S^{\rm opt}\geq1-p_0\cos^2\theta.}
$$
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