= Solution
Let $A=V^{1/2}$ be the symmetric positive square root of the <covariance matrix>, and write $S_1=\mu+AZ$ with $Z$ having the <multivariate normal distribution> $N_d(0,I)$. Bounded gradient makes $g$ globally Lipschitz, so $g(S_1)$ has at most linear growth and is square integrable. This also justifies the permitted <Gaussian integration by parts> formula for each coordinate.
By the <chain rule>, $\nabla_z g(\mu+Az)=A^T\nabla g(\mu+Az)$. Applying Gaussian integration by parts gives
$$
c=\mathbb E[(S_1-\mu)g(S_1)]
=A\,\mathbb E[Zg(\mu+AZ)]
=AA^T\mathbb E[\nabla g(S_1)]
=V\mathbb E[\nabla g(S_1)].
$$
Thus the <Gaussian quadratic hedge> holds $\pi^*=\mathbb E[\nabla g(S_1)]$. Substitution into its initial capital formula yields
$$
\boxed{X_0^*=\frac1{1+r}\mathbb E[g(S_1)]
+\left(S_0-\frac\mu{1+r}\right)\cdot\mathbb E[\nabla g(S_1)].}
$$
Only the hedge calculation is being used here; the Gaussian model need not possess a positive pricing weight for this least-squares identity.
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