Conducting steady states of doped bilayer graphene have a nonzero sublattice pseudospin polarization. Electron-electron interactions renormalize this polarization even at zero temperature, when the phase space for electron-electron scattering vanishes. We show that, because of the strength of interlayer tunneling, electron-electron interactions nevertheless have a negligible influence on the conductivity, which vanishes as the carrier number density goes to zero. The influence of interactions is qualitatively weaker than in the comparable cases of single-layer graphene or topological insulators, because the momentum-space layer pseudospin vorticity is 2 rather than 1. Our study relies on the quantum Liouville equation in the first Born approximation with respect to the scattering potential, with electron-electron interactions taken into account self-consistently in the Hartree-Fock approximation and screening in the random phase approximation. Within this framework the result we obtain is exact.