We explore, for the first time in lattice QCD, the radiative decays to $\gamma \eta$ and $\gamma \eta \prime$ of the $J^{PC}=1^{+-}$ charmonium state, $h_c$. This work expands upon a previous calculation of $J/\psi$ decays to the same final states, incorporating novel technology to access the $h_c$ where it is an excited state and to separate the independent electric dipole and longitudinal form–factors. Results for the radiative decay rates, computed on three-flavor lattices with $m_{} $ 391 MeV show a suppression relative to experiment that is similar to that observed in the prior calculation of $J/\psi \to \gamma {\eta }^{(\prime )}$. The timelike $Q^2$ dependence of both form–factors together influence the Dalitz decays, $h_c\to {\ell }^{+}{\ell }^{-}{\eta }^{(\prime )}$, which are as yet unmeasured experimentally.