= Solution
Once a mode is well inside the horizon, radiation pressure opposes gravitational collapse. For an ideal radiation fluid its <cosmological sound speed> is $c_s=1/\sqrt3$, and its density perturbation undergoes <subhorizon radiation acoustic oscillations> with frequency approximately $k/\sqrt3$. Unlike the superhorizon growing mode, its amplitude does not acquire sustained gravitational growth. Free-streaming relativistic components can additionally phase-mix the perturbation; the acoustic-fluid description is the appropriate idealization for the stated system.
During deep <radiation domination>, the radiation source in the dark-matter equation oscillates rapidly compared with a Hubble time. Its accumulated late-time effect averages away. More quantitatively, an oscillatory source of magnitude $\Delta_r/\tau^2$ gives a rapidly varying particular solution of order $\Delta_r/(k\tau)^2$ when $k\tau\gg1$. It can therefore be neglected in computing the slowly varying homogeneous dark-matter growth, although its forcing near horizon entry fixes the integration constants.
The dark-matter self-gravity term is also subleading while $a/a_{\rm eq}\ll1$. Keeping the leading expansion drag and averaging the radiation source yields
$$
\Delta_c''+\frac1\tau\Delta_c'\simeq0,\qquad (\tau\Delta_c')'\simeq0.
$$
Consequently
$$
\boxed{\Delta_c=C_1+C_2\ln(\tau/\tau_h),\qquad \Delta_{c,\rm grow}\propto\ln\tau.}
$$
This <logarithmic growth of matter perturbations during radiation domination> is the <Mészáros effect>. It is the leading behavior well after horizon entry but before equality; close to equality the neglected self-gravity matters. The radiation perturbation is not small simply because it is radiation: its rapid oscillations, rather than a small radiation background density, justify dropping its source for this slow mode.
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