We consider light scattering by a coated magnetoplasmonic nanoparticle with a Kerr-type nonlinear plasmonic shell and a magneto-optic core. Such a structure features two plasmon dipole modes, associated with electronic oscillations on the inner and outer surfaces of the shell. Driven in a nonlinear regime, each mode exhibits a bistable response. Bistability of an inner plasmon leads to switching between this state and a Fano resonance (Fano switching). Once the external light intensity exceeds a critical value, the bistability zones of both eigenmodes overlap, yielding optical tristability characterized by three stable steady states for a given wavelength and light intensity. We develop a dynamic theory of transitions between nonlinear steady states and estimate the characteristic switching time to be as short as 0.5 ps. We also show that the magneto-optical effect allows red and blue spectral shifts of the Fano profile for right and left circular polarizations of the external light, rendering Fano switching sensitive to light polarization. Specifically, one can reach Fano switching for the right circular polarization while cancelling it for the left circular polarization. The results point to a class of ultrafast Fano switchers tunable by a magnetic field for applications in nanophotonics.