We demonstrate the existence of a cross Feshbach resonance by strongly driving a lower polariton mode and by monitoring in time the transmission of a short optical pulse at the energy of the upper polariton mode in a semiconductor microcavity. From the signatures of the optical resonance, strength, and sign of the energy shift, we attribute the origin of the scattering process between polariton modes with opposite circular polarization to a biexciton bound state. From this study, we infer the conditions required for a strong enhancement of the generation of entangled photon pairs.