I'm sorry but he deserves a full refund. You got scammed and didn not know the item you where selling was not a PE. I can understand your frustration but it is not fair to your client because you did claim to own a PE. The fact that the client did not know what a PE looks like exactly is not his fault since you as a seller claimed the item was in fact a PE. The fact that you didn't know the item was not a PE is neither here or there at this point. It is the responsibility of the seller to make sure that the item he is selling is in fact genuine.
I do feel for both you guys though, this was fucked up.
Space Gray 2011 335xi E92