= Solution
For pressureless matter, $\Delta_M=\delta-3\Psi$ obeys $\Delta_M'=-\theta$. The Euler equation gives
$$
\Delta_M''+\mathcal H\Delta_M'=-k^2\Psi.
$$
The constraint is now
$$
\Psi=-\frac{3\mathcal H^2}{2k^2+9\mathcal H^2}\left(\Delta_M-\frac{3\mathcal H}{k^2}\Delta_M'\right).
$$
Consequently
$$
\boxed{\Delta_M''+(1+\mathcal G)\mathcal H\Delta_M'-\frac13\mathcal Gk^2\Delta_M=0,\qquad
\mathcal G=\frac{9\mathcal H^2}{2k^2+9\mathcal H^2}}.
$$
For a matter-dominated flat background, $a\propto\eta^2$ and $\mathcal H=2/\eta$. On small scales $\mathcal G\ll1$ but $\mathcal Gk^2\to9\mathcal H^2/2$: the last term must not be dropped with $\mathcal G$. The equation becomes
$$
\Delta_M''+\frac2\eta\Delta_M'-\frac6{\eta^2}\Delta_M=0,
\qquad \Delta_M=C\eta^2+D\eta^{-3}.
$$
Since $\Psi$ is small relative to the growing density term on these scales, this recovers $\boxed{\delta_{\rm grow}\propto\eta^2\propto a}$.
There is a printed error in the requested super-horizon decaying mode. At $k\ll\mathcal H$, $\mathcal G\to1$ and the leading equation is $\Delta_M''+4\Delta_M'/\eta=0$, hence
$$
\boxed{\Delta_M=\alpha+\beta\eta^{-3}\quad\text{to leading super-horizon order}}.
$$
The printed $\alpha+\beta/\eta$ fails its own equation: for $\Delta_M=1/\eta$, the left side is $-2/\eta^3\ne0$ at leading order. The $1/\eta$ decaying solution would instead result from using the radiation-era background $\mathcal H=1/\eta$ in this limiting damping term. Finite-$k$ corrections restore terms of relative order $(k\eta)^2$ and do not repair the wrong leading power.
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