I believe that the heat generated by a piston engine would require extra drag for cooling. Also the simple fact of the usage of jet fuel simplifies a turbine/centrifugal compressor type APU simplifies turbine APU usage, supplying an alternate fuel source would be impracticable.
Not to mention the regulated air source for a piston engine that would require a supercharger for apu's that ate certified to operate at altitude. The other factor is creating a piston powered apu that could maintain the 400 hz required for the generator and reliable power to do so.