My understanding has been essentially your second bullet: the 12V keeps the computer happy while the car is parked, and the main battery only kicks in if climate controls are activated or the 12V runs down to the point where it needs to be charged. Once the latter happens, the big battery charges the little battery, then wash/rinse/repeat.
Not the case??
I believe it is not the case for your car (though mine is an older one so I don't have personal experience of it). If you dig around on TMC, somewhere you will find a battery teardown of a more recent pack showing the low-power 12V output to maintain the computers.