The intensity of the Rayleigh line is proportional to

The Rayleigh scattered intensity is proportional to the square of the induced (oscillating) dipole moment, $I \propto \omega^4 \mu_{\mathrm{ind}}^2$ with $\mu_{\mathrm{ind}} = \alpha E$. It is therefore not proportional to the square root of the dipole moment or of the polarizability, nor to the inverse of the polarizability.