= Solution
The nonrelativistic fluid assumptions give
$$
T_{00}=\rho+O(\epsilon^2\rho),\qquad
T_{0i}=O(\epsilon\rho),\qquad
T_{ij}=O(\epsilon^2\rho).
$$
Assume again a stationary field and asymptotic flatness. The harmonic-gauge <Linearized Einstein equations> become
$$
\nabla^2\bar h_{00}=-16\pi\rho,
$$
while the other trace-reversed components are smaller. Trace reversal then gives $h_{00}=\bar h_{00}/2$ to leading order, so
$$
\nabla^2h_{00}=-8\pi\rho.
$$
Using $h_{00}=-2\Phi$ yields
$$
\boxed{\nabla^2\Phi=4\pi\rho}
$$
in units $G=1$, which is the <Poisson equation> of Newtonian gravity.
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