= Solution
For the <synchronous-to-Newtonian scalar Fourier transformation>, the coefficient of $\widehat k_i\widehat k_j$ in the transformed spatial <metric perturbation> must vanish. Thus $h_s+2kL=0$. Vanishing shift then gives $L'=kT$. For $k\ne0$ these conditions fix
$$
\boxed{L=-\frac{h_s}{2k},\qquad T=-\frac{h_s'}{2k^2},\qquad
\xi^\mu=\left(-\frac{h_s'}{2k^2},-\frac{i\widehat k_i h_s}{2k}\right)e^{i\mathbf k\cdot\mathbf x}.}
$$
The lapse in <conformal Newtonian gauge> is $h_{00}^{\rm N}=-2\Phi$, so
$$
\Phi=-(T'+\mathcal HT)=\boxed{\frac{h_s''+\mathcal Hh_s'}{2k^2}}.
$$
The remaining coefficient of $\delta_{ij}$ is
$$
\frac{h-h_s}{3}-2\mathcal HT=-2\Psi,
$$
giving
$$
\boxed{\Psi=-\frac{h-h_s}{6}-\frac{\mathcal Hh_s'}{2k^2}.}
$$
The homogeneous $k=0$ perturbation requires separate background-coordinate choices; the inverse powers of $k$ do not define a transformation for it.
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