THE GUY I BOUGHT the car from removed the license plate. I drove home w/o one. I was passed at one point by a police car, so it’s clearly not a big thing. Unfortunately, the guy kept the screws that hold the plate in place. I stopped at a car parts place on the way to register the car a couple days later. The screws were the wrong size. I stopped at a different place. Google to the rescue.
A few days later I went through the car wash. The license plate was half off the car! Closing the lift gate or going through a car wash was enough to cause the left half of the plate holder to fall free. This became the first repair.
This was a five-minute job including washing the part of the lift gate covered by the license plate. The license plate is attached to a frame with four screws. The frame is attached to the lift gate with two screws that go into plastic plugs. The left most plug was worn/damaged enough not to stay snapped into the sheet metal. Perhaps this has been a problem for BMW, the new part is a big chunkier than the old one. I reinstalled the frame and reattached the license plate. Now it’s time to hit the carwash!
The part is called a “plug-in nut” and has BMW part number 51188242693.