= Solution
Using the conventional critical scale rather than guessing a precise Fourier-width coefficient,
$$
\boxed{E_c=\hbar\omega_c=\frac{3\hbar c\gamma^3}{2R}
=1.07\times10^2\ \mathrm{eV},\qquad
\lambda_c=\frac{hc}{E_c}\simeq11.6\ \mathrm{nm}.}
$$
This is extreme ultraviolet and the very soft X-ray boundary, with a broad spectrum extending on either side; it is not a hard, multi-keV X-ray source at the stated large bending radius. The characteristic <energy> is not a sharp upper cutoff. For orientation, the standard angle-integrated spectral <power> per unit <frequency> peaks near $0.29E_c\simeq31$ eV, also in the extreme ultraviolet; a particular angular acceptance weights that spectrum differently.
The crucial scale follows from both the $3$ GeV <energy> and the $561$ m radius. Using only the fact that synchrotron sources can make X-rays, or replacing the bending radius by a much smaller unstated value, would misidentify the characteristic <photon> energies here. The critical-energy convention and the broadband character are described by the primary radiation model.<Synchrotron Radiation in the Geant4 Physics Reference Manual by Geant4 Collaboration>[photon-spectrum section, equation (209), and Properties of the <Power> Spectra]{ref}
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