OK, success. Yes, the first one was tough. After getting the plastic covers off, I now understand how to do this the easiest.

The plastic cover piece's rounded nose end has a lip that catches on the edge of the handle mounting plate that is bolted (Phillips head screw) into the car chassis.

I found that pushing the cover away from the center of the handle provides the least amount of resistance in popping the cover off of the mounting plate. Pinching the cover also pushes the rounded nose a little more. The hardest part is preventing the flat head screwdriver from scratching the plastic panel.