Several small Polycyclic Aromatic Hydrocarbons (PAHs) have been identified recently in the Taurus Molecular Cloud (TMC-1) using radio telescope observations. Reproducing the observed abundances of these molecules has been a challenge for astrochemical models. Rapid radiative cooling of PAHs by Recurrent Fluorescence (RF), the emission of optical photons from thermally populated electronically excited states, has been shown to efficiently stabilize small PAHs following ionization, augmenting their resilience in astronomical environments and helping to rationalize their observed high abundances. Here, we use a novel method to experimentally determine the radiative cooling rate of the cation of 1-cyanonaphthalene (C10H7CN, 1-CNN), the neutral species of which has been identified in TMC-1. Laser-induced dissociation rates and kinetic energy release distributions of 1-CNN cations isolated in a cryogenic electrostatic ion-beam storage ring are analysed to track the time evolution of the vibrational energy distribution of the initially hot ion ensemble as it cools. The measured cooling rate is in good agreement with the previously calculated RF rate coefficient. Improved measurements and models of the RF mechanism are needed to interpret astronomical observations and refine predictions of the stabilities of interstellar PAHs.