The first three-dimensional simulations of a high power 0.351 mum laser beam propagating through a high temperature hohlraum plasma are reported. We show that 3D fluid-based modeling of stimulated Brillouin scattering, including linear kinetic corrections, reproduces quantitatively the experimental measurements, provided it is coupled to detailed hydrodynamics simulation and a realistic description of the laser beam from its millimeter-size envelope down to the micron scale speckles. These simulations accurately predict the strong reduction of stimulated Brillouin scattering measured when polarization smoothing is used.