A traffic stop isn't the time to explain to the cop that you are right and they are wrong. You can try to do it then, but you might spend the entire day and evening with your new friends.
If you refuse to carry ID, they can detain you if they don't buy your verbal identification. And, well, even if they're in the wrong, your butt is going in the squad car, isn't it?
I always carry ID and a credit card. I want to be sure I can get a pint at any pub I happen by.