We propose a Bell-measurement node for quantum repeaters based on a planar semiconductor microcavity operating in the strong-coupling regime. Cavity photons hybridize with quantum-well excitons to form polaritons combining properties of photons and matter quasiparticles. A control photon loaded into one polariton mode changes the polarization response seen by a subsequently incident target photon. This conditional rotation is governed by the interplay of self-induced Larmor precession triggered by spin-dependent exciton-exciton interactions and the polarization beats caused by the splitting of transverse-electric and transverse-magnetic cavity modes. We identify conditions of the experiment that enable implementation of a controlled-Z gate and allow to distinguish all four Bell states in the ideal limit. The one-sided scattering scheme provides a lower interaction threshold than a scalar Kerr reference under the same assumptions. At a selected operating point, the bandwidth-induced identification error scales as the fourth power of the pulse bandwidth in the narrow-band limit. An additional fixed input rotation reduces this error even further. We describe the entanglement swapping between remote memories and determine the minimum quality of the elementary links needed to obtain an entangled output.