= Solution
For a spatial index $j$, the background <Christoffel symbols> give
$$
\bar\Gamma^\mu{}_{\mu0}=3H,
\qquad
-\bar\Gamma^i{}_{0j}\delta T^0{}_i=-H\delta T^0{}_j,
\qquad
-\bar\Gamma^0{}_{ij}\delta T^i{}_0=-a^2H\delta T^j{}_0.
$$
Together with the $3H\delta T^0{}_j$ trace term, the first two contributions combine to $2H\delta T^0{}_j$. Varying the <Levi-Civita connection> and contracting it with $\bar T^\mu{}_{\nu}=\operatorname{diag}(-\bar\rho,\bar P,\bar P,\bar P)$ gives
$$
\delta\Gamma^\mu{}_{\mu j}=\frac12\partial_j(h^\mu{}_{\mu}),
$$
and the surviving combination reduces to
$$
-(\bar\rho+\bar P)
\left(\frac12\partial_jh_{00}-Hh_{j0}\right).
$$
Consequently the spatial momentum equation is
$$
\boxed{
\partial_0\delta T^0{}_j+\partial_i\delta T^i{}_j
+2H\delta T^0{}_j-a^2H\delta T^j{}_0
-(\bar\rho+\bar P)
\left(\frac12\partial_jh_{00}-Hh_{j0}\right)=0}.
$$
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