Any time I have occasion to speak to law enforcement officers who seem curious about such things, my standard response is, "I prefer not to talk about such things." Bill Clinton's clever turn of phrase, "don't ask, don't tell" applies. Whenever the conversation turns to ownership, possession, or use of firearms, keep in mind the five fingers of your right hand. Look at those fingers and consider that the five fingers represent five consonants: "K Y B M S". (Keep Your Big Mouth Shut). If threatened with or placed under arrest, the only thing you should say is, "I want my lawyer." Keep in mind that no one has to advise you of your rights unless and until you be placed in full custodial arrest, so the usual practice is to interrogate "suspects" before putting them in handcuffs. KYBMS.
If a LEO demands, note I said, "demands", to see your CHP certificate (the permission itself is merely a license, which is by definition intangible - the card is evidence of permission), that's the time you have to display the card and in no other situation. I use my U.S. Passport as the required government issued picture I.D., by the way, unless the cop already has my O.L. because it's a traffic stop. (I don't travel, so I really got the passport so as to get into the U.S. District Courthouses.) If the cop has to ask, or make some expression about what he "needs", that can safely be ignored. Don't let them use the politeness your mom taught you against you - police interaction is not a social call, no matter how sociable they appear to (and may really) be. They would not be talking to you unless The State was considering whether you're a Bad Guy who needs to be punished.
KYBMS. Here endeth the lesson.