Nonlinear Josephson circuits play a crucial role in the growing landscape of quantum information and technologies. The typical circuits studied in this field consist of qubits, whose anharmonicity is much larger than their linewidth, and also of parametric amplifiers, which are engineered with linewidths of tens of MHz or more. The regime of small anharmonicity but also narrow linewidth, corresponding to the dynamics of a high-Q Duffing oscillator, has not been extensively explored using Josephson cavities. Here, we use two-tone spectroscopy to study the susceptibility of a strongly driven high-Q Josephson microwave cavity. Under blue-detuned driving, we observe a shift of the cavity susceptibility, analogous to the AC Stark effect in atomic physics. When applying a strong red-detuned drive, we observe the appearance of an additional idler mode above the bifurcation threshold with net external gain. Strong driving of the circuit leads to the appearance of two exceptional points and a level attraction between the quasimodes of the driven cavity. Our results provide insights on the physics of driven nonlinear Josephson resonators and form a starting point for exploring topological physics in strongly-driven Kerr oscillators.