Since the Lazarsfeld-Mukai bundle $E = E_{C,A}$ fits into the exact sequence

$$0 \to H^0(C,A)^\vee \otimes \mathcal{O}_S \to E \xrightarrow{\phi} K_C(-A) \to 0,$$

the first Chern class of $E$ is  $[C]$. Let $F$ be a subsheaf of $E$ and let $K = \ker(\phi_{|F})$ and $L= \mathrm{im}(\phi_{|F})$.  Since $K$ and $L$ are subsheaves of $V :=H^0(C,A)^\vee \otimes \mathcal{O}_S$ and $K_C(-A)$ respectively, one has

$$c_1(F) \cdot [C] = (c_1(K) +c_1(L))\cdot [C] \le [C]\cdot [C] = c_1(E) \cdot [C].$$

The equality holds when $L = K_C(-A)$ and $K$ is a direct sum of copies of $\mathcal{O}_S$. In this case, the quotient $E/F \simeq V/K$ is also a  direct sum of copies  of $\mathcal{O}_S$. As $H^0(E^\vee) = 0$, we conclude that $E = F$.

Therefore if $F$ is a proper saturated subsheaf of $E$ such that $\mu_C(F) \ge \mu_C(E)$, we would have
$$[C]\cdot [C] > c_1(F)\cdot [C] \ge \left(1-\frac{rkF}{rkE}\right)[C]\cdot [C] > 0,$$
which is in contradiction to the assumption that $Pic(S) = \mathbf{Z} \cdot[C]$.