A spin-photon interface based on the luminescence of a singly charged quantum dot in a micropillar cavity allows for the creation of photonic entangled states. Current devices suffer from cavity birefringence, which limits the generation of spin-photon entanglement. In this paper, we theoretically study the light absorption and emission by the interface with an anisotropic cavity and derive the maximal excitation and spin-photon entanglement conditions. We show that the concurrence of the spin-photon state equal to one and complete quantum dot population inversion can be reached for a micropillar cavity with any degree of birefringence by tuning the quantum dot resonance strictly between the cavity modes. This sweet spot is also valid for generating a multiphoton cluster state, as we demonstrate by calculating the three-tangle and fidelity with the maximally entangled state.