I suppose if the DL check does show one is the holder of a CCW, and the officer asks the driver about it, it would make good sense for the driver to reveal any weapons he is carrying concealed.
CA law does permit any officer encountering a firearm to inspect it and run a check to verify the weapon is not stolen. If the owner and possessor of the weapon does not cooperate, then the owner - possessor can be arrested.
If the CCW holder has cooperated with the officer and produced his permit and weapon for inspection, it would be a stretch of the imagination for the officer to come up with probable cause that the CCW holder is concealing other illegal weapons. And provided the officer has not arrested the driver and CCW holder, the officer does not have authority to search the areas of the car immediately accessible to the driver at the time of the traffic stop.
So if the officer wants to search the car he is going to have to come up with probable cause to search the car if the driver and CCW holder does not give permission to search. Said probable cause will have to withstand judicial scrutiny as if the officer had requested a warrant to search the car.
Judicial scrutiny of a warrantless search will examine the "totality of the circumstances" that existed at the time which caused the officer to believe that searching the premises will result in discovery of seizable items. In the scenario you describe, the CCW holder is cooperating with every request by the officer regarding the existence of a permit and a weapon. The court will ask the officer what did the driver do that caused him to think he was concealing illegal weapons or contraband. If the officer's only reason is the existence of a CCW, I think a court would rule, "Sorry, that is not an indicator of illegal behavior."
It was my experience that crooks trying to hide something are overly cooperative in other areas and volunteer information that a normal person would keep his mouth shut about. Therefore, I suggest that if the law does not require it, don't volunteer information that is not asked for. Keep your hands on the steering wheel where the nice officer can see them. If he learns about the permit and asks about it, then tell him the information he desires, like where the weapon is located.
I could not tell from the original post if Skunkabilly has a CCW permit or not. If he does not and is going through the initial 16 hour training, question the instructor on anything he says he purports to be the law. A good instructor will back up anything he says with written handouts that refer to sections in the penal code or enabling regulations published by the Department of Justice.
In the same vein, Skunkabilly, if you have a CCW and return to the four hour renewal class, ask for documentation on any "changes" in the law that have occurred since the last time you were issued a permit. It will be good insurance to keep urban legends from becoming "CCW law."