IMO, using a car door as a shield, you are not in the category of "intentionally displays the handgun in plain view of another person in a public place". It doesn't seem you are "displaying" it to anyone in this scenario, simply moving it from holster to safe. One can definitely argue that you are intentionally displaying it, but IMO, you would get the officer (unless one of *those* anti-gun) and/or courts to agree you weren't intentionally displaying. Using the car door, car, and yourself to shield it, I can't think that is "plain view" unless people are already just standing on all sides of your vehicle watching you
Of course, having a truck is much nicer for disarming at the door than a small car.
That being said, simple solution for peace of mind - keep a small towel near your safe. Cover your hand/ weapon as you draw it and it is definitely not in plain view of anyone.