Employing the moving particles' semi-implicit (MPS) method, this study presents a numerical framework for solving the Navier-Stokes equations for the propagation and the split of a liquid plug through a three-dimensional air-filled bifurcating tube, where the inner surface is coated by a thin fluid film, and surface tension acts on the air-liquid interface. The detailed derivation of a modified MPS method to handle the air-liquid interface of liquid plugs is presented. When the front air-liquid interface of the plug splits at the bifurcation, the interface deforms quickly and causes large wall shear stress. We observe that the presence of a transverse gravitational force causes asymmetries in plug splitting, which becomes more pronounced as the capillary number decreases or the Bond number increases. We also observe that there exists a critical capillary number below which the plug does not split into two daughter tubes but propagates into the lower daughter tube only. In order to deliver the plug into the upper daughter tube, the driving pressure to push the plug is required to overcome the hydrostatic pressure due to gravity. These tendencies agree with our previous experimental and theoretical studies.