= Solution
The massless scalar <stress tensor> in mixed-index form is
$$
T^\mu{}_{\nu}=g^{\mu\alpha}\partial_\alpha\phi\,\partial_\nu\phi
-\frac12\delta^\mu{}_{\nu}g^{\alpha\beta}\partial_\alpha\phi\,\partial_\beta\phi.
$$
In <conformal Newtonian gauge>, $g^{00}=-a^{-2}(1-2\Phi)$ and $g^{ij}=a^{-2}(1+2\Psi)\delta^{ij}$ to first order. Spatial <gradients> of the background vanish, so products of two spatial perturbation <gradients> are second order. Consequently
$$
g^{\alpha\beta}\partial_\alpha\phi\,\partial_\beta\phi
=-a^{-2}(\phi_0'^2+2\phi_0'\delta\phi'-2\Phi\phi_0'^2).
$$
Substitution yields
$$
\boxed{a^2T^0{}_{0}=-\frac12\phi_0'^2-\phi_0'\delta\phi'+\Phi\phi_0'^2,
\qquad a^2T^0{}_{i}=-\phi_0'\partial_i\delta\phi.}
$$
The potential $\Psi$ multiplies only second-order <gradient> products in these components. The same reasoning makes the off-diagonal spatial stress vanish at first order, consistent with $\Phi=\Psi$.
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