Okay, some additional information.
I only got one key set with the car.
The local GM dealership checked to see if I could obtain a duplicate, and they can do it for me.
There was a second fob with no keys on it (photo).
I tested the first fob with the keys, and it opens my hatch.
The other did nothing.
I opened the second FOB case to install a new battery and noticed that it required some minor soldering.
It had a good battery, but it just wasn't connected.
So I went back to it to see if it worked and noticed that snowflake symbol from the switch inside the car on that fob! Lo and behold, it says Clifford.
That matches up with a
Clifford G4 Immobilizer security system.
So the mystery is solved, although I am searching for the installation and accurate user manuals.
I have a 10-page G4/G5 SmartWindows User Guide.
That provides some information, but otherwise lacks detail.
But apparently, other model car owners have been saddled with this system.
I should be able to track something down.
It is amazing how much paper they can waste to distribute next to nothing useful.
The installation was very well done, and it made me think it was installed at the factory.
SLP might have done it.