We introduce a new model for the accretion and feedback of supermassive black hole (SMBH) binaries to the ketjucode, 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 on to the secondary SMBH, which is motivated by results from small-scale hydrodynamical circumbinary disc simulations. We perform idealized gas-rich disc galaxy merger simulations using pure thermal or pure kinetic active galactic nuclei (AGNs) 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(merge) similar to 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.