= Solution
The supplied matter-era solution has $h'=h_S'=Ak^{1/2}\tau/2$, so the source reduces to $Ak^{1/2}\mu^2\tau$. Put $D=\tau_0-\tau_*$ and $W=e^{-ik\mu D}$. The integrating factor for the streaming equation gives the exact <matter-era photon endpoint solution>
$$
\boxed{\Delta_0=W\Delta_*+Ak^{1/2}\mu^2\int_{\tau_*}^{\tau_0}\tau e^{-ik\mu(\tau_0-\tau)}d\tau.}
$$
For $b=k\mu\ne0$, the integral is $(\tau_0-\tau_*W)/(ib)+(1-W)/b^2$. Substitution and collection of endpoint terms gives
$$
\boxed{\Delta_0=
W\left[\Delta_*+iAk^{-1/2}\mu\tau_*-Ak^{-3/2}\right]
+Ak^{-3/2}-iAk^{-1/2}\mu\tau_0.}
$$
At $\mu=0$ the original source is zero and the expression has the continuous limit $\Delta_0=\Delta_*$. The integral form is numerically preferable near that limit, where the separated endpoint terms cancel.
The last two terms are respectively a local observer monopole and dipole. For measured higher angular multipoles, remove them. The remaining emission bracket contains initial brightness, an emission velocity-type term, and a gravitational <Sachs-Wolfe effect> term. Under ordinary <adiabatic initial conditions> for the growing synchronous matter mode, $\Delta_*\sim Ak^{1/2}\tau_*^2$, while the velocity endpoint scales as $Ak^{-1/2}\tau_*$. Relative to $Ak^{-3/2}$ these are suppressed by $(k\tau_*)^2$ and $k\tau_*$ respectively. Thus for modes outside the horizon at last scattering,
$$
\boxed{\Delta_{\ell\ge2}\ \text{is dominated by the emission term}\ -Ak^{-3/2}e^{-ik\mu D}.}
$$
Its $k^{-3/2}$ amplitude is the scale-invariant large-angle contribution. In Newtonian language it is the ordinary endpoint <Sachs-Wolfe effect>, not a late integrated effect in a constant-potential matter universe. With completely arbitrary prescribed $\Delta_*$, no unconditional dominance theorem is possible: an independently large initial anisotropy could dominate. The exact solution above remains valid without that adiabatic assumption.
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