Due to the fact that Doppelgangers deal energy damage and are as stated in the wiki an apparitions, it means they are an energy based manifestation.
So it makes sense that they are immune to physical damag, but energy weapons should harm them or cause a status, where if you hit them with enough energy they implode because of too much energy in the concentated energy field that keeps them in the world.
This could also cause an energy backlash on the person creating them as they technically have to be connected.