= Solution
Choose the incident photon along the $x$ direction. Conservation of <four-momentum> preserves the squared total energy invariant $\mathcal S=E_{\rm tot}^2-c^2|\mathbf p_{\rm tot}|^2$. Initially,
$$
\mathcal S=(mc^2+E)^2-E^2=m^2c^4+2mc^2E.
$$
In the centre-of-momentum frame let the outgoing photon energy be $e_*>0$; the larger particle has opposite momentum magnitude $e_*/c$ and energy $\sqrt{M^2c^4+e_*^2}$. Thus
$$
\sqrt{\mathcal S}=e_*+\sqrt{M^2c^4+e_*^2}>Mc^2.
$$
Conversely, whenever $\mathcal S>M^2c^4$, the positive value $e_*=(\mathcal S-M^2c^4)/(2\sqrt{\mathcal S})$ constructs a final state with the required invariant, which can be boosted to the laboratory frame. The limiting <photon excitation threshold with a soft final photon> is therefore
$$
\boxed{E_{\rm th}=\frac{M^2-m^2}{2m}c^2.}
$$
Strictly, an outgoing physical photon requires $E>E_{\rm th}$; the printed minimum is the threshold infimum approached as the final photon becomes soft.
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