You are not required in any state, by law, to carry proof of reciprocity. You are only required to possess a valid permit. Exactly how is showing the officer a website that says your permit is good legal proof anyway? It's just a website or phone app, exactly what legal weight does that carry?
He was looking for you to show him a reason not to write you a ticket that he had absolutely no idea would hold up in court or not, as a matter of his convenience, not legal requirement. I am really surprised to read this about Washington State Patrol - they are usually much more professional than that.
Anyone who says, "I support the 2nd amendment, BUT"... doesn't. Element of Surprise: a mythical element that many believe has the same affect upon criminals that Kryptonite has upon Superman.