Radiation pressure does not have anything to do with mass-energy equivalence. 1), the energy for this process does not come from the conversion of electric energy into mass, and 2) having a momentum is not a property tied to massive particles. All electromagnetic waves carry momentum according to Maxwell’s theory of electromagnetism, and its transfer is fully explained by his equations and results from the interaction of the EM wave with matter, i.e., absorption and reflection. Each such interaction will transfer momentum to the massive object. This is classical physics, you don’t need any Einstein relativity to explain electromagnetic phenomena because his theories only become relevant for very massive bodies and their movement, and nuclear reactions.