The extraction of decay parameters using lattice techniques is a computationally expensive task, requiring several volumes and group irreps to relate the spectrum on a lattice simulation to the infinite volume scattering. In this project we employ an alternative method based on a narrow-width approximation to extract the hadronic mixing $<\bar{D}D|\psi(3770)>$, which is needed to compute the decay $\Gamma(\psi(3770)\to\bar{D}D)$ between the second excited state of vector charmonium and a pair of $D$-mesons in a $p$-wave. We carry out our lattice simulations on two CLS ensembles at $m_\pi \sim 440~\text{MeV}$ and $a\sim 0.066~\text{fm}$ and obtain results compatible with experiment. Furthermore, we interpret our results analytically using the ${}^3P_0$ quark model.