= Solution
To match the positive source printed in the question, choose <synchronous gauge> metric
$$
ds^2=a^2\left[-d\tau^2+(\delta_{ij}-h_{ij})dx^idx^j\right].
$$
The commonly used convention with $\delta_{ij}+h_{ij}$ gives the opposite source sign; switching conventions means replacing $h$ and $h_S$ by their negatives together. This distinction is necessary when comparing with the usual <synchronous photon brightness equation>.
Let $p^\alpha=dx^\alpha/d\lambda$ be the photon tangent for an affine parameter. The covariant <geodesic equation> is
$$
\frac{d\widetilde p_\nu}{d\lambda}=\frac12g_{\alpha\beta,\nu}p^\alpha p^\beta.
$$
For $\nu=0$, the homogeneous scale-factor contributions cancel by the <null geodesic> condition, leaving $d\widetilde p_0/d\lambda=-a^2h'_{ij}p^ip^j/2$. Since $\widetilde p_0=-a^2p^0=-q$ and $d\tau/d\lambda=p^0$, use the background $p^i=p^0n^i$ in this first-order term to obtain
$$
\boxed{q'=\frac12q h'_{ij}n^in^j.}
$$
Using the affine form also avoids confusing a covariant $p_0$ with the contravariant factor $p^0$ required when changing the geodesic parameter to coordinate time.
In <conformal time>, the equation from part(a) becomes $f_1'+n^i\partial_i f_1+q'f_0'=0$. A Fourier mode proportional to $e^{i\mathbf k\cdot\mathbf x}$ has streaming term $ik\mu f_1$. Multiply by $-4/(qf_0')$; its prefactor is background-time independent. With $\Delta=-4f_1/(qf_0')$,
$$
\Delta'+ik\mu\Delta=4q'/q=2h'_{ij}n^in^j.
$$
Contracting the scalar decomposition explicitly gives
$$
h'_{ij}n^in^j=\frac{h'}3+\left(\mu^2-\frac13\right)h_S',
$$
so
$$
\boxed{\Delta'+ik\mu\Delta=\frac23(h'-h_S')+2\mu^2h_S'.}
$$
This is the <photon brightness equation for a negative spatial perturbation>. For a perturbation of blackbody-temperature form, $f_1=-qf_0'\Theta$, so $\Delta=4\Theta$ is the <photon brightness perturbation>. A more general initial spectral distortion can retain $q$ dependence; the gravitational forcing is still independent of $q$ in this linear massless equation.
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