We investigate induced Compton/Brillouin scattering of electromagnetic waves in magnetized electron and positron pair plasma by verifying kinetic theory with Particle-in-Cell simulations. Applying this to fast radio bursts (FRBs) in magnetar magnetospheres, we find that the scattering–although suppressed by the magnetic field–inevitably enters the linear growth stage before the incident wave amplitude becomes comparable to the background magnetic field during its outward propagation through the magnetosphere. The subsequent evolution bifurcates: full scattering occurs when the density exceeds a critical value, whereas below it the scattering saturates and the FRB can escape without substantial induced-scattering attenuation. This nonlinear saturation eases the tension with observations of compact emission regions and may explain the observed diversity, including the presence or absence of FRBs associated with X-ray bursts.