$$
\sum_{i=1}^n (Y_i-\bar Y)^2 \sim \sigma^2\chi^2_{n-1}.
$$
"Everybody knows" that $\mathbb E(\chi^2_k)=k$. What is perhaps less widely known is that $\operatorname{var}(\chi^2_k)=2k$. Where that comes from I will say something about below.
That implies
$$
\operatorname{var}\left( c\sum_{i=1}^k (Y_i-\bar Y)^2 \right) = c^2\sigma^4 2(n-1).
$$
So
\begin{align}
& \mathbb E\left(\left( \sigma^2- c\sum_{i=1}^n(Y_i-\bar Y)^2 \right)^2\right) \\[10pt]
= & \sigma^4 -2\sigma^2\Big(c(n-1)\sigma^2\Big) + c^2\mathbb E((\chi^2_{n-1})^2) \\[10pt]
= & \sigma^4 -2\sigma^2\Big(c(n-1)\sigma^2\Big) + c^2 \sigma^4\Big( \operatorname{var}(\chi^2_{n-1}) + (n-1)^2 \Big). \tag 1
\end{align}
If I have the details right, the value of $c$ that makes $(1)$ as small as possible should be $1/(n+1)$ (thus an estimator with even more bias than the MLE is "best" in the mean-squared-error sense).
So where did we get $\operatorname{var}(\chi^2_k)= 2k\ {}$?
Since the chi-square random variable as distributed as $Z_1^2+\cdots+Z_k^2$ where the $Z$s are i.i.d. standard normals, the variance should be $k\operatorname{var}(Z_1^2)$. We have
$$
\operatorname{var}(Z^2) = \mathbb E(Z^4) - \left(\mathbb E(Z^2)\right)^2 = \mathbb E(Z^4) - 1. \tag 2
$$
This is
\begin{align}
& \int_{-\infty}^\infty z^4 \varphi(z)\,dz = 2\int_0^\infty z^4 \varphi(z)\,dz \\[10pt]
= & 2\frac{1}{\sqrt{2\pi}} \int_0^\infty z^4 e^{-z^2/2} \, dz = 2\frac{1}{\sqrt{2\pi}} \int_0^\infty z^3 e^{-z^2/2} \Big( z \, dz \Big) \\[10pt]
= & 2\frac{1}{\sqrt{2\pi}} \int_0^\infty \sqrt{2}^3 u^{3/2} e^{-u} \, du = 2\frac{1}{\sqrt{2\pi}} \sqrt{2}^3 \Gamma\left(\frac 5 2 \right) \\[10pt]
= & 2\frac{1}{\sqrt{2\pi}} \cdot \sqrt{2}^3\cdot \frac 3 2 \cdot \frac 1 2 \Gamma\left(\frac 1 2 \right)
\end{align}
Recalling that $\Gamma(1/2)=\sqrt{\pi}$, the above reduces to $3$, so in view of $(2)$, we have $\operatorname{var}(\chi^2_1)=2$.
A postscript prompted by comments below: We had
$$
\sigma^4 -2\sigma^2\Big(c(n-1)\sigma^2\Big) + c^2 \sigma^4\Big( \operatorname{var}(\chi^2_{n-1}) + (n-1)^2 \Big). \tag 1
$$
Then we found that $\operatorname{var}(\chi^2_{n-1})=2(n-1)$, so $(1)$ becomes
$$
\sigma^4 -2\sigma^2\Big(c(n-1)\sigma^2\Big) + c^2 \sigma^4\Big(2(n-1) + (n-1)^2 \Big)
$$
This is $\sigma^4$ times
$$
1 -2 \Big(c(n-1) \Big) + c^2 \Big(2(n-1) + (n-1)^2 \Big).
$$
By routine algebra this becomes
$$
g(c) = (n-1)(n+1)c^2 -2(n-1)c + 1.
$$
As a function of $c$, this is a parabola that opens upward, so the lowest point is the one value of $c$ at which $g'(c)=0$. So find
$$
g'(c) = 2(n-1)(n+1) c - 2(n-1) = 2(n-1)\Big( (n+1)c - 1 \Big).
$$
Clearly the expression in the $\Big(\text{big parentheses}\Big)$ is $0$ precisely when $c = \dfrac{1}{n+1}$.
(It can also be done by completing the square.)
