= Solution
Use $\mathcal H=a'/a$ and the Fourier convention $\nabla\to i\mathbf k$. A valid linearization requires all components of $h_{ij}$ to be small: the printed small-determinant condition alone is insufficient, as the matrix $\operatorname{diag}(1,1,0)$ has zero determinant without being small. This is <small determinant does not imply a small metric perturbation>.
Define the <density contrast> $\delta=(\rho-\bar\rho)/\bar\rho$ and the <scalar velocity potential> by $v^i=\partial^i\theta$, or $v^i_{\mathbf k}=ik^i\theta_{\mathbf k}$. Hence $\nabla\cdot\mathbf v=-k^2\theta$. To first order, $u^0=a^{-1}$, $u^i=a^{-1}v^i$ and $g^{ij}=a^{-2}(\delta^{ij}-h^{ij})$. The product $u^iu^j$ is second order. Substitution into the <perfect fluid> <stress-energy tensor> gives
$$
\boxed{T^{00}=a^{-2}\bar\rho(1+\delta),\quad
T^{0i}=a^{-2}(1+w)\bar\rho\,ik^i\theta,\quad
T^{ij}=a^{-2}w\bar\rho[(1+\delta)\delta^{ij}-h^{ij}].}
$$
Here the equation of state is used locally, so the pressure perturbation is $w\bar\rho\delta$.
The contraction in $\nabla_\mu T^{0\mu}=0$ is
$$
\partial_\mu T^{0\mu}+\Gamma^0_{\mu\nu}T^{\nu\mu}+\Gamma^\mu_{\mu\nu}T^{0\nu}=0.
$$
Using the displayed connections gives, through first order,
$$
\rho'+3\mathcal H(\rho+P)+(\bar\rho+\bar P)\left(\nabla\cdot\mathbf v+\frac12h'\right)=0.
$$
In particular $\Gamma^\mu_{\mu0}=4\mathcal H+h'/2$; the spatial connection contraction adds $3\mathcal HP+\bar P h'/2$, while differentiating $a^{-2}\rho$ supplies $-2\mathcal H\rho$. Subtract the background relation $\bar\rho'+3\mathcal H(1+w)\bar\rho=0$. The background terms multiplying $\delta$ cancel, leaving
$$
\boxed{\delta'-(1+w)k^2\theta+\frac12(1+w)h'=0.}
$$
For comoving <cold dark matter>, $w=0$ and $\theta=0$, so $\delta_c'=-h'/2$ and $\delta_c+h/2$ is constant at each spatial point. The proper-volume element of a comoving cell is $a^3\sqrt{\det(I+h)}\,d^3x=a^3(1+h/2)d^3x$. Conserved mass in the cell then gives the density change opposite to the perturbation of its volume. \b[In comoving synchronous coordinates the linear CDM density change is a volume change, not transport across cell boundaries.]
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