= Solution
The linear <density contrast> is Gaussian, so its cubic <moment> vanishes. The leading contribution is
$$
\langle\delta^3\rangle=3\langle(\delta^{(1)})^2\delta^{(2)}\rangle+\cdots.
$$
Two connected <Wick contractions> give
$$
\langle\delta^3\rangle
=6D^4\int\frac{d^3q_1d^3q_2}{(2\pi)^6}
P_{\rm lin}(q_1)P_{\rm lin}(q_2)F_{2,s}(\mathbf q_1,\mathbf q_2).
$$
The third pairing vanishes because $F_{2,s}(\mathbf q,-\mathbf q)=0$. Isotropy makes $\langle\mu\rangle=0$ and $\langle\mu^2\rangle=1/3$, so the angular average of the kernel is
$$
\frac57+\frac27\frac13=\frac{17}{21}.
$$
Since $\sigma^2=D^2\int d^3q\,P_{\rm lin}(q)/(2\pi)^3$ to leading order, the <unsmoothed matter-density reduced skewness> is
$$
\boxed{\langle\delta^3\rangle=\frac{34}{7}\sigma^4+O(\sigma^6),
\qquad S_3=\frac{34}{7}+O(\sigma^2)}.
$$
This assumes finite unsmoothed <variance>; inserting a smoothing window generally changes $S_3$.
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