= Solution
Let $Q^{ij}=\hat k^i\hat k^j-\delta^{ij}/3$. Rotations around $\hat{\mathbf k}$ and the vanishing trace imply that the angular integral in the <scalar neutrino anisotropic stress> has the form $A_\ell Q^{ij}$. Contract with $\hat k_i\hat k_j$ and use $\mu^2-1/3=(2/3)P_2(\mu)$. By the <Orthogonality of Legendre polynomials>,
$$
\frac23 A_\ell=\frac12\int_{-1}^{1}P_\ell(\mu)\left(\mu^2-\frac13\right)d\mu
=\frac23\frac{\delta_{\ell2}}5,
\qquad A_\ell=\frac{\delta_{\ell2}}5.
$$
Only the quadrupole in the <neutrino Boltzmann hierarchy> contributes. Its phase factor is $(-i)^2=-1$, so the supplied stress definition becomes
$$
\Pi^{\hat i\hat j}(\eta,\mathbf x)
=\frac45\bar\rho_\nu\int\frac{d^3\mathbf k}{(2\pi)^{3/2}}\Theta_2(\eta,\mathbf k)Q^{ij}e^{i\mathbf k\cdot\mathbf x}.
$$
Comparing with the chosen normalization $-(4/3)\bar\rho_\nu\Pi Q^{ij}$ gives
$$
\boxed{\Pi(\eta,\mathbf k)=-\frac35\Theta_2(\eta,\mathbf k)}.
$$
The sign and factor here follow jointly from the stress convention and the unweighted <Legendre polynomial> coefficients; they should not be transferred unchanged to a differently normalized hierarchy.
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