This paper proposes a computational solver via the localized radial basis function finite difference (RBF-FD) scheme and the use of graded meshes for solving the time-dependent Bates partial integro-differential equation (PIDE) arising in computational finance. In order to avoid facing a large system of discretization systems, we employ graded meshes along both of the spatial variables, which results in constructing a set of ordinary differential equations (ODEs) of lower sizes. Moreover, an explicit time integrator is used because it can bypass the need to solve the large discretized linear systems in each time level. The stability of the numerical method is discussed in detail based on the eigenvalues of the system matrix. Finally, numerical tests revealed the accuracy and reliability of the presented solver.