We present measurements on a superconducting single-electron transistor (SET) in which the metallic tip of a low-temperature scanning force microscope is used as a movable gate. We characterize the SET through charge stability diagram measurements and compare them to scanning gate measurements taken in the normal conducting and the superconducting states. The tip-induced potential is found to have a rather complex shape. It consists of a gate voltage-dependent part and a part which is independent of gate voltage. Further scanning gate measurements reveal a dependence of the charging energy and the superconducting gap on the tip position and the voltage applied to it. We observe an unexpected correlation between the magnitude of the superconducting gap and the charging energy. The change in EC can be understood to be due to screening, however the origin of the observed variation in Δ remains to be understood. Simulations of the electrostatic problem are in reasonable agreement with the measured capacitances.