The magnetoresistance of a three-dimensional Rashba metal placed on top of a ferromagnetic insulator is theoretically investigated. In addition to the intrinsic Rashba spin-orbit interaction, we consider extrinsic spin-orbit coupling via side-jump and skew scattering, as well as Elliott-Yafet spin relaxation. The latter is anisotropic due to the mass anisotropy which reflects the noncentrosymmetric crystal structure. A quasiclassical approach is employed to derive coupled spin-diffusion equations, which are supplemented by boundary conditions that account for the spin-transfer torque at the interface of the bilayer. The magnetoresistance is fully determined by the current-induced spin polarization, i.e., it cannot in general be ascribed to a single (bulk) spin Hall angle. Our theory reproduces several features of the experiments, at least qualitatively, and contains established phenomenological results in the relevant limiting cases. In particular, the anisotropy of the Elliott-Yafet spin relaxation mechanism plays a major role for the interpretation of the observed magnetoresistance.