We explore the nonlinear response to a strong drive of polaritonic meters for superconducting qubit state readout. The two polaritonic meters result from the strong hybridization betweena bosonic mode of a 3D microwave cavity and an anharmonic ancilla mode of the superconducting circuit. Both polaritons inherit a self-Kerr nonlinearity U, and decay rate κ from the ancilla and cavity, respectively. They are coupled to a transmon qubit via a non-perturbative cross-Kerr coupling resulting in a large cavity pull 2χ>κ, U. By applying magnitic flux, the ancilla mode frequency varies modifying the hybridization conditions and thus the properties of the readout polariton modes. Using this, the hybridisation is tuned in the mesoscopic regime of the non-linear dissipative polariton where the self-Kerr and decay rates of one polariton are similar U∼κ leading to bistability and bifurcation behavior at small photon number. This bistability and bifurcation behavior depends on the qubit state and we report qubit state readout in a latching-like manner thanks to the bifurcation of the upper polariton. Without any external quantum-limited amplifier, we obtain a single-shot fidelity of 98.6% in a 500 ns integration time.