= Solution
Let the medium's <autocorrelation function of a random field> and <power spectrum> use the explicit convention
$$
C(\mathbf s)=\langle W(\mathbf r+\mathbf s)W(\mathbf r)\rangle,\qquad
S_W(\mathbf q)=\int_{\mathbb R^3}C(\mathbf s)e^{-i\mathbf q\cdot\mathbf s}\,d^3s,
$$
$$
C(\mathbf s)=\frac1{(2\pi)^3}\int_{\mathbb R^3}S_W(\mathbf q)e^{i\mathbf q\cdot\mathbf s}\,d^3q.
$$
Since $n^2-1=2\mu W+\mu^2W^2$, only the linear term in $\mu$ is needed to obtain the cross-section through second order:
$$
f_\infty^{(1)}=\frac{\mu k^2}{2\pi}\int_VW(\mathbf r)e^{-i\mathbf k_s\cdot\mathbf r}\,d^3r+O(\mu^2).
$$
Squaring and taking the <expectation> gives
$$
\boxed{\langle\sigma_d\rangle=
\frac{\mu^2k^4}{4\pi^2}\int_V\!\int_V
C(\mathbf r-\mathbf r')e^{-i\mathbf k_s\cdot(\mathbf r-\mathbf r')}\,d^3r\,d^3r'
+O(\mu^3).}
$$
This is the finite-volume formula in the <Born approximation for scalar wave scattering>; the zero mean of $W$ does not make its squared scattered amplitude vanish. For a jointly centered <Gaussian random field>, the third-order contribution is zero, so the next perturbative contribution is fourth order when that expansion is valid.
Introduce the overlap volume $H_V(\mathbf s)=|V\cap(V-\mathbf s)|$. Changing variables rewrites the double integral as
$$
\int_{\mathbb R^3}H_V(\mathbf s)C(\mathbf s)e^{-i\mathbf k_s\cdot\mathbf s}\,d^3s.
$$
When the linear dimensions of a regular scattering volume are large compared with the <correlation length>, and $C$ is integrable, $H_V(\mathbf s)\simeq|V|$ over the significant covariance range. This gives the <finite-volume random scattering spectrum> result
$$
\boxed{\langle\sigma_d(\widehat{\mathbf r},\widehat{\mathbf r}_0)\rangle
\simeq\frac{\mu^2k^4|V|}{4\pi^2}S_W(\mathbf k_s),\qquad
\mathbf k_s=k(\widehat{\mathbf r}-\widehat{\mathbf r}_0).}
$$
There are two distinct approximations here: the weak-scattering expansion in $\mu$ and the negligible-boundary-overlap approximation in the volume-to-correlation-scale ratio. If $S$ instead denotes the <power spectrum> of the index fluctuation $n-1=\mu W$, then $S=\mu^2S_W$ and the external factor $\mu^2$ is absorbed into $S$.
<Statistical homogeneity> alone allows an anisotropic spectrum, so its argument is generally the vector $\mathbf k_s$. If <statistical isotropy> is additionally assumed, $S_W$ depends only on
$$
\boxed{k_s=|\mathbf k_s|=2k\sin(\theta/2),\qquad
\cos\theta=\widehat{\mathbf r}\cdot\widehat{\mathbf r}_0.}
$$
The scalar notation $S(k_s)$ is valid in that isotropic case. For the alternative convention $S_{\mathrm{norm}}=(2\pi)^{-3}S_W$, the bulk formula is $2\pi\mu^2k^4|V|S_{\mathrm{norm}}(\mathbf k_s)$. Changing the sign of the scattering vector does not change the spectrum of a real stationary scalar field; changing the <Fourier transform> normalization does change the displayed $2\pi$ factors. Both conventions have therefore been stated explicitly.
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