As far as I’m concerned, a cop doesn’t need to know that I have a gun unless he gives me a reason for him to be looking down the barrel. Hold your cards close.
That said, I would show my permit if asked, or when showing my normal ID. But I don’t think it’s right. It’s none of his business. Just like my MMA status is none of his business.
When he runs your plate if it is your car he will already know your status. It’s best to hand over your permit with your license so it does not look like you are trying to hide something
When I took my class, the instructor recommended that if asked if there is a gun in the vehicle to first respond that you’re a licensed conceal carry permit holder.