In this paper from year $2016$ is given the more general approximation
$$\mathcal{S}_n^{(m)}=\frac 1{2 \sqrt{\pi }}\,\frac{n! \,\left(e^R-1\right)^m }{R^n\, m!\, \sqrt{m H R}}\,\,\left(1+O\left(\frac 1 {m^{1/3} } \right)\right)$$ where
$$R=k+W\left(-k\,e^{-k}\right) \qquad \text{and} \qquad H=\frac{k \left(1+W\left(-k\,e^{-k} \right)\right)}{2 \left(k+W\left(-k\,e^{-k}
\right)\right)}\qquad \text{and} \qquad k=\frac n m$$
Using it for the specific case
$$\mathcal{S}_{2 n}^{(n)}\sim \frac{(2 n)! \left(-W\left(-\frac{2}{e^2}\right)
\left(2+W\left(-\frac{2}{e^2}\right)\right)\right)^{-n}}{2 \sqrt{\pi n}\, n! \sqrt{
1+ W\left(-\frac{2}{e^2}\right)} } \tag1$$
while the asymptotic formula from Vaclav Kotesovec write
$$\mathcal{S}_{2 n}^{(n)}\sim \frac{2^{2 n-\frac{1}{2}} n^{n-\frac{1}{2}} \left(-e W\left(-\frac{2}{e^2}\right)
\left(W\left(-\frac{2}{e^2}\right)+2\right)\right)^{-n}}{\sqrt \pi \sqrt{1+
W\left(-\frac{2}{e^2}\right)}}\tag 2$$
Trying to improve the approximation, a purely empirical approach gives
$$\mathcal{S}_{2 n}^{(n)}\sim \frac{(2 n)! \left(-W\left(-\frac{2}{e^2}\right)
\left(2+W\left(-\frac{2}{e^2}\right)\right)\right)^{-n}}{2 \sqrt{\pi n}\, n! \sqrt{
1+ W\left(-\frac{2}{e^2}\right)} } \left(1-\frac{1}{\pi \sqrt{\pi } \left(2+W\left(-\frac{2}{e^2}\right)\right)
n}\right)\tag3$$
Computing their logarithms, it seems that $(1)$ is slightly better than $(2)$.
$$\left(
\begin{array}{ccccc}
n & \text{from }(2) & \text{from }(1) & \text{from }(3) & \text{exact} \\
5 & 10.6881 & 10.6798 & 10.6570 & 10.6578 \\
10 & 29.4240 & 29.4198 & 29.4085 & 29.4089 \\
15 & 50.9200 & 50.9172 & 50.9096 & 50.9100 \\
20 & 74.1738 & 74.1717 & 74.1661 & 74.1663 \\
25 & 98.7233 & 98.7216 & 98.7171 & 98.7173 \\
30 & 124.300 & 124.299 & 124.295 & 124.295 \\
35 & 150.728 & 150.727 & 150.724 & 150.724 \\
40 & 177.883 & 177.882 & 177.879 & 177.879 \\
45 & 205.673 & 205.672 & 205.669 & 205.669 \\
50 & 234.025 & 234.024 & 234.022 & 234.022
\end{array}
\right)$$
In fact $$\frac{(1)}{(2)}=\frac{e^n \Gamma \left(n+\frac{1}{2}\right)}{\sqrt{2 \pi }\,n^n}=1-\frac{1}{24 n}+\frac{1}{1152 n^2}+O\left(\frac{1}{n^3}\right)$$