= Solution
Divide the time component of the <geodesic equation> by $(P^0)^2$ and use $d/d\lambda=P^0d/d\eta$. To first order, $P^i/P^0=\widehat p^i(1+\delta N-\widehat{\mathbf p}\mathbin\cdot\nabla\psi)$. Substitution of the supplied <Christoffel symbol>[Christoffel symbols] makes the background $2\mathcal H$ terms cancel the derivative of $a^{-2}$. The derivatives of the lapse and shift also combine, leaving
$$
\boxed{\frac{\epsilon'}\epsilon
=-\widehat p^i\partial_i(\delta N+\psi')}.
$$
Thus the comoving photon energy is conserved in the unperturbed spacetime and changes through the gradient of the scalar gravitational source.
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