= Binding scale and recoil
A distinct classical <Thomson scattering> window requires two separated scales:
$$
\omega_0\ll\omega\ll\omega_C,\qquad \omega_C=\frac{mc^2}{\hbar}.
$$
To remove that window, the binding frequency must be comparable with or greater than the <Compton scattering> recoil scale. Thus the formal condition for <bound-electron scattering without a Thomson frequency window> is
$$
\sqrt{\frac km}\gtrsim\frac{mc^2}{\hbar},\qquad
\boxed{k\gtrsim\frac{m^3c^4}{\hbar^2}.}
$$
Using the stipulated approximate <electron mass> $m=10^{-30}\,\mathrm{kg}$, $c=2.998\times10^8\,\mathrm{m\,s^{-1}}$ and $\hbar=1.055\times10^{-34}\,\mathrm{J\,s}$ gives
$$
\boxed{k_{\rm scale}\simeq7.3\times10^{11}\,\mathrm{N\,m^{-1}},\qquad
\omega_C\simeq8.5\times10^{20}\,\mathrm{s^{-1}}.}
$$
The dimensions are $[m^3c^4/\hbar^2]=\mathrm{kg\,s^{-2}}=\mathrm{N\,m^{-1}}$. The inequality denotes a scale comparison, not a uniquely specified spring constant or a sharp transition.
There is also a physical limitation to the question's premise. <Compton scattering> does not begin discontinuously at $\hbar\omega=mc^2$; recoil corrections grow as photon <energy> approaches the rest <energy>. If $\hbar\omega_0\gtrsim mc^2$, the assumed nonrelativistic binding model is itself no longer reliable. The calculation therefore locates where no well-separated classical <Thomson scattering> range exists; it cannot establish a literal direct transition of a real strongly bound system using this oscillator equation alone.
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