= Solution
The four contributions have distinct physical origins. The intrinsic <photon temperature perturbation> is $\delta_\gamma/4$: a local <blackbody radiation> <energy density> satisfies $\rho_\gamma\propto T^4$, so its first-order fractional perturbation is four times the fractional <temperature> perturbation. The endpoint <cosmological Newtonian potential> term $\psi_*$ is the <gravitational redshift> between the emitting plasma and the observer; escaping a negative potential well lowers the measured <photon> <energy>. Together with the intrinsic contribution it produces the ordinary <Sachs-Wolfe effect>. The term $-\hat{\mathbf n}\cdot\mathbf v_*$ is the <Doppler CMB anisotropy>, caused by the plasma's line-of-sight <peculiar velocity> at last scattering, with the source's direction convention.
Finally, $2\int d\chi\,\psi'$ is the <Integrated Sachs-Wolfe effect>. A <photon> passing through a static potential well gains <energy> on entry and loses it on exit, with no net shift. When the potential evolves while the <photon> crosses, the two shifts fail to cancel. The factor two is the sum of the temporal and spatial scalar-potential contributions when <anisotropic stress> is negligible, so the two potentials are equal. The prime is a <partial derivative> at fixed <comoving coordinate>; it is not the total <derivative> along the light ray.
For a low-<cosmological redshift> galaxy sample, the intrinsic, endpoint-potential and last-scattering <Doppler CMB anisotropy> terms are localized near $z\simeq1100$, whereas the <galaxy density contrast> is measured much nearer us. In the <Limber approximation> their radial kernels have no overlap, so their leading cross correlations vanish. The late <Integrated Sachs-Wolfe effect> samples the same nearby evolving potentials as the <galaxies>, giving the desired <late-time CMB-matter cross correlation>. The early <Integrated Sachs-Wolfe effect> near <matter-radiation equality> is likewise radially separated.
This is the leading linear, separated-kernel description intended here. The primary <Cosmic microwave background anisotropy> and late <density contrast> share primordial initial conditions, so their exact large-angle correlation need not be identically zero outside the <Limber approximation>. Additional secondary effects, such as the <Sunyaev-Zeldovich effect>, can also correlate with <galaxies> in real maps; they are absent from the stipulated linear expression.
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