  In this study of the `Resolving supermAssive Black hole Binaries In galacTic
hydrodynamical Simulations' (RABBITS) series, we investigate the orbital
evolution of supermassive black holes (SMBHs) during galaxy mergers. We
simulate both disc and elliptical galaxy mergers using the KETJU code, which
can simultaneously follow galaxy (hydro-)dynamics and small-scale SMBH dynamics
with post-Newtonian corrections. With our SMBH binary subgrid model, we show
how active galactic nuclei (AGNs) feedback affects galaxy properties and SMBH
coalescence. We find that simulations without AGN feedback exhibit excessive
star formation, resulting in merger remnants that deviate from observed
properties. Kinetic AGN feedback proves more effective than thermal AGN
feedback in expelling gas from the centre and quenching star formation. The
different central galaxy properties, which are a result of distinct AGN
feedback models, lead to varying rates of SMBH orbital decay. In the dynamical
friction phase, galaxies with higher star formation and higher SMBH masses
possess denser centres, become more resistant to tidal stripping, experience
greater dynamical friction, and consequently form SMBH binaries earlier. As AGN
feedback reduces gas densities in the centres, dynamical friction by stars
dominates over gas. In the SMBH hardening phase, compared to elliptical
mergers, disc mergers exhibit higher central densities of newly formed stars,
resulting in accelerated SMBH hardening and shorter merger time-scales (i.e.
$\lesssim 500$ Myr versus $\gtrsim 1$ Gyr). Our findings highlight the
importance of AGN feedback and its numerical implementation in understanding
the SMBH coalescing process, a key focus for low-frequency gravitational wave
observatories.
