We develop a theory for quantum phases and quantum multicriticality in bilayer graphene in the presence of an explicit energy gap in the noninteracting spectrum by extending previous renormalization group (RG) analyses of electron-electron interactions in gapless bilayer graphene at finite temperature to include the effect of an electric field applied perpendicular to the sample, which produces an energy gap in the single-particle electron-hole dispersion. We determine the possible outcomes of the resulting RG equations, represented by "fixed rays" along which ratios of the coupling constants remain constant and map out the leading instabilities of the system for an interaction of the form of a Coulomb interaction that is screened by two parallel conducting plates placed equidistant from the electron. We find that some of the fixed rays on the "target plane" found in the zero-field case are no longer valid fixed rays, but that all four of the isolated rays are still valid. We also find five additional fixed rays that are not present in the zero-field case. We then construct maps of the leading instability (or instabilities) of the system for the screened Coulomb-like interaction as a function of the overall interaction strength and interaction range for four values of the applied electric field. We find that the pattern of leading instabilities is the same as that found in the zero-field case, namely, that the system is unstable to a layer antiferromagnetic state for short-ranged interactions, to a nematic state for long-ranged interactions, and to both for intermediate-ranged interactions. However, if the interaction becomes too long ranged or too weak, then the system will exhibit no instabilities. The ranges at which the nematic instability first appears, the antiferromagnetic instability disappears, and the nematic instability disappears all decrease with increasing applied electric field. Our main qualitative finding, that the applied electric field opposes the emergence of symmetry-broken phases in general, suppressing, however, the antiferromagnetc phase more strongly compared with the nematic phase, is directly testable experimentally.