We propose an experiment to find evidence of the formation of light-induced crossings provoked by cavity quantum radiation on simple molecules by using state-of-the-art optical cavities, molecular beams, pump-probe laser schemes, and velocity mapping detectors for fragmentation. The procedure is based on prompt excitation and subsequent dissociation in a three-state scheme of a polar diatomic molecule, with two $^{1}\mathrm{\ensuremath{\Sigma}}$ states (ground and first excited) coupled first by the UV pump laser and then by the cavity radiation, and a third fully dissociative state $^{1}\mathrm{\ensuremath{\Pi}}$ coupled through the delayed UV/V probe laser. The observed enhancement of photodissociation yields in the $^{1}\mathrm{\ensuremath{\Pi}}$ channel at given time delays between the pump and probe lasers unambiguously indicates the formation of a light-induced crossing between the two $^{1}\mathrm{\ensuremath{\Sigma}}$ field-dressed potential energy curves of the molecule. Also, the production of cavity photons out of the vacuum field state via nonadiabatic effects represents a showcase of a molecular dynamical Casimir effect. To simulate the experiment outcome, we perform ab initio coherent quantum dynamics of the molecule LiF subject to external lasers and quantum cavity interactions in the strong coupling regime, using a product grid representation of the total polaritonic wave function for both vibrational and photon degrees of freedom.