The steady-state distribution of a bunched electron beam in a resonator with a large-amplitude carrier signal is obtained self-consistently and for arbitrary injected distribution. It is then employed to analyze the stability of operation for free electron laser (FEL) oscillators fed by a continuous electron beam. The dielectric response to small perturbations (sidebands) includes the nonuniformity from the aperiodic beam bunching and the cross-coupling among symmetric upper and lower sidebands caused by the unharmonic (nonlinear) nature of the trapped particle trajectories. The dominant contribution comes from resonances of the sideband frequency detuning δ=‖ωs−ω0‖ with harmonics of the Doppler-upshifted synchrotron (bounce) frequency ωb, δ≂2γ2znωb. The instability is triggered by the gradients, both across, ∂F/∂J, and around, ∂F/∂θ, the bunched beam distribution F(J,θ), where the action J parametrizes the distance from, and θ, the angle around, the center of the trapped particle island. When the wiggler length becomes much longer than the synchrotron period, the contribution from the nonuniformity around θ phase mixes to zero; one then recovers the results for a FEL amplifier [Phys. Fluids 31, 1708 (1988)]. It is also shown that when the nonlinear shift of the carrier frequency is considerable, upper and lower sidebands decouple, because symmetric sideband frequencies correspond to asymmetric wave numbers.