= Solution
Let $p^0=d\tau/d\lambda$. A comoving observer measures photon energy $p=a p^0$, so $q=ap=a^2p^0$. The null condition is $(\delta_{ij}+h_{ij})p^ip^j=(p^0)^2$. Using the stated <Christoffel symbols> in the time <geodesic> equation gives
$$
\frac{dp^0}{d\lambda}=-2\mathcal H(p^0)^2-\frac12h'_{ij}p^ip^j.
$$
Divide by $p^0$ to differentiate with respect to <conformal time>. The background redshift term cancels when differentiating $a^2p^0$:
$$
q'=a^2\left(\frac{dp^0}{d\tau}+2\mathcal Hp^0\right)
=-\frac{a^2}{2p^0}h'_{ij}p^ip^j.
$$
In the perturbation term use the background relation $p^i/p^0=\widehat n^i$, giving
$$
\boxed{q'=-\frac12q h'_{ij}\widehat n^i\widehat n^j.}
$$
Without <metric tensor> perturbations, both $q$ and the propagation direction are constant along an unperturbed ray. Thus $q'$ and $d\widehat n^i/d\tau$ are first order. Multiplying them by <derivatives> of $f_1$, which is also first order, gives second-order terms. The term $q'df_0/dq$ is first order and must be retained. At linear order the <Collisionless Boltzmann equation> therefore reads
$$
\frac{\partial f_1}{\partial\tau}+\widehat n^i\partial_i f_1
=\frac12q\frac{df_0}{dq}h'_{ij}\widehat n^i\widehat n^j.
$$
Taking the <Fourier transform> in the spatial coordinates proves
$$
\boxed{f_1'+ik\mu f_1=\frac12qf_0'(q)h'_{ij}\widehat n^i\widehat n^j,
\qquad\mu=\widehat{\mathbf k}\cdot\widehat{\mathbf n}.}
$$
A first-order correction to the propagation velocity multiplying a spatial <derivative> of $f_1$ is likewise second order.
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