Motivated by experiments on the carrier-doped bilayer iridate (Sr1-xLax)3Ir2O7, we study the dynamics of a single doped electron in a bilayer magnet in the presence of spin-orbit coupling, taking into account the spatially staggered rotation of IrO6 octahedra. We employ an effective single-orbital bilayer t-J model, concentrating on the quantum paramagnetic phase near the magnetic quantum critical point. We determine the carrier dispersion using a combination of self-consistent Born and bond-operator techniques. Extrapolating to finite small carrier density we find that, for experimentally relevant parameters, the combination of octahedral rotation and spin-orbit coupling induces a band folding which results in a Fermi surface of small double electron pockets, in striking agreement with experimental observations. We also determine the influence of spin-orbit coupling on the location of the quantum critical point in the undoped case, and discuss aspects of the global phase diagram of doped bilayer Mott insulators.