The internal structure of the $X(3872)$ is still a highly debated topic in heavy-quarkonium physics. Here, we investigate its radiative transitions, specifically $X(3872)\to J/\psi\,\gamma$ and $X(3872)\to \psi^\prime\gamma$, assuming a compact tetraquark configuration. The decay amplitudes are evaluated within a non-relativistic approach, relying on the initial tetraquark wave function. To model this state, we apply the Born--Oppenheimer approximation, separating the dynamics of the slow heavy charm quarks from the fast light quarks. By variationally determining the effective Born--Oppenheimer potential, we solve the Schr\"odinger equation for the $c\bar c$ subsystem. Our theoretical prediction for the ratio of the radiative branching fractions is $\mathcal{R}_{\mathrm{th}} = 1.4 \pm 0.3$, which nicely matches recent LHCb data. More broadly, this technique successfully maps out the mass spectrum of compact $c\bar c q\bar q$ configurations, demonstrating the effectiveness of the Born--Oppenheimer method for exploring exotic hadron structures.