Complex soft matter systems can be efficiently studied with the help of adaptive resolution simulation methods, concurrently employing two levels of resolution in different regions of the simulation domain. The nonmatching properties of high- and low-resolution models, however, lead to thermodynamic imbalances between the system's subdomains. Such inhomogeneities can be healed by appropriate compensation forces, whose calculation requires nontrivial iterative procedures. In this work we employ the recently developed Hamiltonian adaptive resolution simulation method to perform Monte Carlo simulations of a binary mixture, and propose an efficient scheme, based on Kirkwood thermodynamic integration, to regulate the thermodynamic balance of multicomponent systems.