How can I get Focus –> Driving –> While Driving –> When Connected to Car Bluetooth" to recognize my car?

In "Settings –> Focus –> Driving –> While Driving," I have it set to "When Connected to Car Bluetooth." But it never notices when my car is connected to Bluetooth. My car connects to Bluetooth as soon as I start it up, and I've identified the BlueTooth type as Car Stereo, but that never seems to activate the driving mode that I specified. Does the "When Connected to Car Bluetooth" even work?

My old Android phone let me specify exactly which bluetooth device was my car. (I had to be connected to the car's Bluetooth first, which was dumb, but at least I could make it work.) My iPhone has no option like that. (I don't have CarPlay, so I can't use that.)

What can I do to get my iPhone 13 Pro to recognize my car's Bluetooth and turn on driving mode?