Bernal-stacked bilayer graphene (BLG), in which a graphene layer is stacked atop another and laterally shifted by a lattice constant, offers remarkable tunability in its single-particle states under applied magnetic and displacement fields. Owing to this tunability, recent transport and scanning tunneling microscopy experiments in the presence of a perpendicular magnetic field and finite interlayer displacement fields have observed even-denominator fractional quantum Hall (FQH) states at half-filling in the first excited, namely, N=2, Landau level (LL) of BLG. In contrast, at zero displacement field, a gapless composite fermion Fermi liquid (CFFL) is realized at half-filling of the N=2 LL. Motivated by these experiments, we compute the phase diagram as a function of the displacement field in the half-filled N=2 LL of BLG by studying the competition between the CFFL and the Moore-Read state—a candidate even-denominator FQH state—by calculating their thermodynamic energies in this setting. We find that the modified effective Coulomb interaction, induced by changes in the single-particle states with increasing displacement field, softens the inter-electronic repulsion at short distances, thereby stabilizing the Moore-Read state over the CFFL in the N=2 LL of BLG. We also study the nature of FQH states at fillings 2/5, 3/7, 4/9, and 6/13 in the N=2 LL of BLG. Our results suggest that, with increasing displacement field, the Jain composite-fermion states at 3/7, 4/9, and 6/13 transition into states with distinct topological order that are well-captured by parton wave functions.