Really all you saying the police will find an excuse to stop you just to see if your carrying.
REALLY COME ON
Yes most states when you run a plate it shows if the REGISTERED owner has a carry permit.
Some states even go as far as listing the guns you have registered to you.
If you are a law abiding citizen and no driving like a jerk there is no reason to run your plate, which leads to no interaction.
If stopped just be yourself, be respectful. If you feel the need to mention you have a permit AND are carrying keep your hands visible and inform the officer where the gun is located and where your wallet is or how ever you carry your IDs. Then ask how the OFFICER would like you to proceed to get the info for him/her.
There's nothing hard about it and no reason to make it hard.