Here is the Wikipedia version of the

Lehmann-Scheffè Theorem Let $\vec{X}= (X_1, X_2, \dots, X_n$) be a random sample from a distribution that has p.d.f (or p.m.f in the discrete case) $f(x:\theta)$ where $\theta \in \Omega$ is a parameter in the parameter space. Suppose $Y = u(\vec{X})$ is a sufficient statistic for $\theta$, and let$$\{ f_Y(y:\theta): \theta \in \Omega\}$$be a complete family. If $\varphi$ is such that $$\operatorname{E}[\varphi(Y)] = \theta$$ then $\varphi(Y)$ is the unique MVUE of $\theta$.

The elements to consider are

1. $S^2(\vec{X})$ is an unbiased estimator of $\text{var}(X_i)=\lambda$.
2. $\bar X(\vec{X})$ is an unbiased estimator of $\operatorname{E}[X_i]=\lambda$.
3. $\bar X(\vec{X})$ is a minimal sufficient complete statistic.
4. $\operatorname{E}[S^2(\vec{X})|\bar X(\vec{X})]$ is both an unbiased estimator of $\operatorname{E}[X_i]=\lambda$ and a function of $\bar X(\vec{X})$.

Here is the Wikipedia version of the

Lehmann-Scheffè Theorem Let $\vec{X}= (X_1, X_2, \dots, X_n$) be a random sample from a distribution that has p.d.f (or p.m.f in the discrete case) $f(x:\theta)$ where $\theta \in \Omega$ is a parameter in the parameter space. Suppose $Y = u(\vec{X})$ is a sufficient statistic for $\theta$, and let$$\{ f_Y(y:\theta): \theta \in \Omega\}$$be a complete family. If $\varphi$ is such that $$\operatorname{E}[\varphi(Y)] = \theta$$ then $\varphi(Y)$ is the unique MVUE of $\theta$.

The elements of relevance to consider when applying this theorem to the setting of the question are

1. $S^2(\vec{X})$ is an unbiased estimator of $\text{var}(X_i)=\lambda$.
2. $\bar X(\vec{X})$ is an unbiased estimator of $\operatorname{E}[X_i]=\lambda$.
3. $\bar X(\vec{X})$ is a sufficient and complete statistic.
4. $\operatorname{E}[S^2(\vec{X})|\bar X(\vec{X})]$ is both an unbiased estimator of $\operatorname{E}[X_i]=\lambda$ and a function of $\bar X(\vec{X})$.

Conclusion follows with no further computation.