Numerical modelling and simulations are conducted on the large-amplitude dynamics of a functionally graded microcantilever with a tip-mass, additionally supported by an intermediate spring; the functionally graded microsystem is subject to a base excitation. Since one end of the microsystem is free to move, it undergoes large deformation; curvature-related nonlinearities play an important role. Taking into account this type of nonlinearity, using the Mori–Tanaka homogenisation scheme, as well as the modified couple stress theory, an energy technique is employed to derive the nonlinearly coupled equations for the longitudinal and transverse motions. An inextensibility assumption is applied for the functionally graded microcantilever, and hence, a nonlinear equation of motion for the transverse motion involving inertial (apart from stiffness) nonlinearity is obtained. For the functionally graded microsystem considered, effects of the length-scale parameter, the material gradient index, the tip-mass, and the stiffness of the spring-support on the nonlinear resonant responses are highlighted by means of a Houbolt’s finite difference scheme together with Newton–Raphson method.