The nature of the $X(3872)$ remains one of the central open problems in heavy-hadron spectroscopy. In this contribution, we study the radiative decays $X(3872)\to J/\psi\,\gamma$ and $X(3872)\to \psi^\prime\gamma$ within a compact tetraquark interpretation. We employ a non-relativistic framework in which the decay amplitude is determined by the initial-state tetraquark wave function. This wave function is constructed using the Born--Oppenheimer approximation, treating the heavy charm quarks as slow degrees of freedom and the light quarks as fast ones. The resulting Born--Oppenheimer potential is obtained variationally and used to solve the effective Schr\"odinger equation for the $c\bar c$ pair. We compute the ratio of radiative branching fractions and obtain $\mathcal{R}_{\mathrm{th}} = 1.4 \pm 0.3$, in good agreement with the recent LHCb measurement. Beyond this observable, the same framework provides a systematic description of the spectrum of compact $c\bar c q\bar q$ states, highlighting the versatility of the Born--Oppenheimer approach in the study of exotic hadrons.