= Solution
Let $\Gamma=-\dot\tau=a\bar n_e\sigma_T$ and let $D=\partial_\eta+\mathbf e\cdot\nabla$ differentiate along the unperturbed photon trajectory. Use $1+(\mathbf e\cdot\hat{\mathbf m})^2=4/3+(2/3)P_2(\mathbf e\cdot\hat{\mathbf m})$ to decompose the <Thomson scattering> gain term:
$$
\frac{3\Gamma}{16\pi}\int d\hat{\mathbf m}\,\Theta(\hat{\mathbf m})[1+(\mathbf e\cdot\hat{\mathbf m})^2]
=\Gamma[\Theta_0+Q(\mathbf e)],
\quad
Q(\mathbf e)=\frac12\int\frac{d\hat{\mathbf m}}{4\pi}\Theta(\hat{\mathbf m})P_2(\mathbf e\cdot\hat{\mathbf m}).
$$
The <Orthogonality of Legendre polynomials> identifies $Q$ as a pure <photon quadrupole>. Dropping $Q$ is an approximation, justified when <tight coupling> makes the scattering quadrupole small; the supplied unpolarized equation already neglects polarization corrections. The <Cosmic microwave background line-of-sight solution> requested here omits that quadrupole source.
Set $Y=\Theta+\psi$. Adding $D\psi=\dot\psi+\mathbf e\cdot\nabla\psi$ to the <Photon Boltzmann equation with Thomson scattering> yields
$$
DY+\Gamma Y\simeq\dot\phi+\dot\psi+\Gamma(\Theta_0+\psi+\mathbf e\cdot\mathbf v_b).
$$
Along $\mathbf x(\eta)=\mathbf x_0-(\eta_0-\eta)\mathbf e$, the integrating factor is $e^{-\tau}$ because $d e^{-\tau}/d\eta=\Gamma e^{-\tau}$. Consequently
$$
\frac{d}{d\eta}[e^{-\tau}Y(\eta,\mathbf x(\eta),\mathbf e)]
\simeq e^{-\tau}(\dot\phi+\dot\psi)+g(\Theta_0+\psi+\mathbf e\cdot\mathbf v_b).
$$
An early optically thick boundary makes $e^{-\tau}Y$ vanish for finite initial perturbations, and $e^{-\tau(\eta_0)}=1$. Therefore
$$
\boxed{\begin{aligned}
\Theta(\eta_0,\mathbf x_0,\mathbf e)+\psi(\eta_0,\mathbf x_0)
&\simeq\int_{\eta_i}^{\eta_0}g(\eta)(\Theta_0+\psi+\mathbf e\cdot\mathbf v_b)(\eta,\mathbf x_0-\chi\mathbf e)d\eta\\
&\quad+\int_{\eta_i}^{\eta_0}e^{-\tau}(\dot\phi+\dot\psi)(\eta,\mathbf x_0-\chi\mathbf e)d\eta,
\end{aligned}}\qquad\chi=\eta_0-\eta.
$$
Here $\dot\phi$ and $\dot\psi$ are partial time derivatives, not total derivatives along the ray. We work at first order in <scalar cosmological perturbations>, evaluate sources along the background <radial null geodesic in FLRW spacetime>, and ignore observer peculiar velocity. Perturbed paths and lensing affect this first-order source solution only at higher perturbative order. A moving observer adds its local Doppler dipole.
The original PDF, unlike the transcribed TeX, also asks for the instantaneous-last-scattering limit and its interpretation. For $g(\eta)=\delta^{(D)}(\eta-\eta_*)$ with no later scattering, and a matter-era growing mode with constant $\phi$ and $\psi$, the <Integrated Sachs-Wolfe effect> vanishes. Hence
$$
\boxed{\Theta_{\rm obs}(\mathbf e)\simeq\Theta_0(\eta_*,\mathbf x_*)+\psi(\eta_*,\mathbf x_*)-\psi(\eta_0,\mathbf x_0)
+\mathbf e\cdot\mathbf v_b(\eta_*,\mathbf x_*),\quad\mathbf x_*=\mathbf x_0-\chi_*\mathbf e}.
$$
The four terms are the intrinsic emission temperature, the gravitational potential at emission, the observer's potential, and the <Doppler CMB anisotropy>. The two potentials describe the gravitational redshift; the observer term is independent of direction and is absorbed into the measured temperature monopole. The intrinsic temperature and emission potential form the <Sachs-Wolfe combination>. On <superhorizon scales> for <adiabatic initial conditions> in <matter domination>, $\Theta_0=-2\psi/3$, giving the ordinary <Sachs-Wolfe effect> $\Theta_0+\psi=\psi/3$. The Doppler sign is positive with the paper's propagation direction $\mathbf e$; the viewing direction is $-\mathbf e$.
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