A method to achieve stable radiation pressure acceleration (RPA) of heavy ions from laser-irradiated ultrathin foils is proposed, where a high-Z material coating in front is used. The coated high-Z material, acting as a moving electron repository, continuously replenishes the accelerating heavy ion foil with comoving electrons in the light-sail acceleration stage due to its successive ionization under laser fields with Gaussian temporal profile. As a result, the detrimental effects such as foil deformation and electron loss induced by the Rayleigh-Taylor-like and other instabilities in RPA are significantly offset and suppressed so that stable acceleration of heavy ions are maintained. Particle-in-cell simulations show that a monoenergetic Al^{13+} beam with peak energy 3.8 GeV and particle number 10^{10} (charge >20 nC) can be obtained at intensity 10^{22} W/cm^{2}.