We study the process of magnetic reconnection in a coalescing magnetic island setup by means of numerical simulation. This process mimics flux tube merging which can take place in the solar corona, laboratory, and astrophysical objects. Simulations are performed with magnetohydrodynamics (MHD), Hall-MHD, and a newly developed Coupled MHD and Particle-In-Cell (PIC) model (CMAP). This model treats the global simulation domain with MHD, while the region around the reconnection zone is treated with PIC. This CMAP code allows us to simulate larger-scale domains with lesser computing power compared to fully PIC simulations. CMAP reproduces the dynamics of fully kinetic simulations which Hall-MHD does not capture, as seen in the Hall magnetic field and the reconnecting current sheet structure. For large islands in kinetic simulations, the current sheet does not form smoothly and shows chaotic behavior, and the magnetic islands also bounce and slosh. The current sheet thickness, length, and aspect ratios are calculated. They show that in the CMAP model, the thickness remains close to the ion skin depth, while the length changes weakly with the system size, giving a steady aspect ratio for the two largest system size simulations. The pressure tensor also shows large deviations from isotropy and gyrotropy near the current sheet. The CMAP simulations for smaller system sizes are compared to fully kinetic simulations, and we find that a minimum fraction of area has to be provided PIC feedback in the CMAP simulations in order to produce reconnection rates and dynamics similar to fully kinetic simulations. The reconnection rate reduces with the increasing island size. For the CMAP model, this reduction is steeper compared to MHD and Hall-MHD initially, but for larger system sizes, the reconnection rates in CMAP simulations show a steady behavior.