  Dynamical encounters of stellar-mass black holes (BHs) in a disc of compact
objects around a supermassive BH (SMBH) can accelerate the formation and
coalescence of BH binaries. It has been proposed that binary-single encounters
among BHs in such discs can lead to an excess of highly-eccentric BH mergers.
However, previous studies have neglected how the disc velocity dispersion and
the SMBH's tidal field affect the 3-body dynamics. We investigate the outcomes
of binary-single encounters considering different values of the disc velocity
dispersion, and examine the role of the SMBH's tidal field. We then demonstrate
how their inclusion affects the properties of merging BH binaries. We perform
simulations of 4-body encounters (i.e. with the SMBH as fourth particle) using
the highly-accurate, regularized code TSUNAMI, which includes post-Newtonian
corrections up to order 3.5PN. The disc velocity dispersion controls how orbits
in the disc are aligned and circular, and determines the relative velocity of
the binary-single pair before the encounter. As the velocity dispersion
decreases, the eccentricity of post-encounter binaries transitions from thermal
to superthermal, and binaries experience enhanced hardening. The transition
between these two regimes happens at disc eccentricities and inclinations of
order e ~ i ~ 10^-4. These distinct regimes correspond to a disc dominated by
random motions, and one dominated by the Keplerian shear. The inclusion of the
SMBH's tidal field and the disc velocity dispersion can significantly affect
the number of GW mergers, and especially the number of highly-eccentric
inspirals. These can be up to ~2 times higher at low velocity dispersion, and
~12 times lower at high velocity dispersions. The spin-orbit alignment is
influenced by the tidal field exclusively at high velocity dispersions,
effectively inhibiting the formation of anti-aligned binary BHs.
