  We introduce a new model for the accretion and feedback of supermassive black
hole (SMBH) binaries to the KETJU code, which enables us to resolve the
evolution of SMBH binaries down to separations of tens of Schwarzschild radii
in gas-rich galaxy mergers. Our subgrid binary accretion model extends the
widely used Bondi--Hoyle--Lyttleton accretion into the binary phase and
incorporates preferential mass accretion onto the secondary SMBH, which is
motivated by results from small-scale hydrodynamical circumbinary disc
simulations. We perform idealised gas-rich disc galaxy merger simulations using
pure thermal or pure kinetic active galactic nuclei (AGN) feedback. Our binary
accretion model provides more physically motivated SMBH mass ratios, which are
one of the key parameters for computing gravitational wave (GW) induced recoil
velocities. The merger time-scales of our simulated SMBH binaries are in the
range $t_{\rm merge}{\sim} 10$--$400$ Myr. Prograde in-plane equal-mass galaxy
mergers lead to the shortest merger time-scales, as they experience the
strongest starbursts, with the ensuing high stellar density resulting in a
rapid SMBH coalescence. Compared to the thermal AGN feedback, the kinetic AGN
feedback predicts longer merger time-scales and results in more core-like
stellar profiles, as it is more effective in removing gas from the galaxy
centre and quenching star formation. This suggests that the AGN feedback
implementation plays a critical role in modelling SMBH coalescences. Our model
will be useful for improving the modelling of SMBH mergers in gas-rich
galaxies, the prime targets for the upcoming LISA GW observatory.
