We assess the contribution of dynamical hardening by direct three-body scattering interactions to the rate of stellar-mass black hole binary (BHB) mergers in galactic nuclei. We derive an analytic model for the single-binary encounter rate in a nucleus with spherical and disc components hosting a super-massive black hole (SMBH). We determine the total number of encounters N_GW needed to harden a BHB to the point that inspiral due to gravitational wave emission occurs before the next three-body scattering event. This is done independently for both the spherical and disc components. Using a Monte Carlo approach, we refine our calculations for N_GW to include gravitational wave emission between scattering events. For astrophysically plausible models, we find that typically N_GW ≲ 10. We find two separate regimes for the efficient dynamical hardening of BHBs: (1) spherical star clusters with high central densities, low-velocity dispersions, and no significant Keplerian component and (2) migration traps in discs around SMBHs lacking any significant spherical stellar component in the vicinity of the migration trap, which is expected due to effective orbital inclination reduction of any spherical population by the disc. We also find a weak correlation between the ratio of the second-order velocity moment to velocity dispersion in galactic nuclei and the rate of BHB mergers, where this ratio is a proxy for the ratio between the rotation- and dispersion-supported components. Because discs enforce planar interactions that are efficient in hardening BHBs, particularly in migration traps, they have high merger rates that can contribute significantly to the rate of BHBmergers detected by the advanced Laser Interferometer Gravitational-Wave Observatory.