12V battery question

Neider of your choices. If the aux battery drained it's power, the vehicle will not start up by it's own means, it will have to be power charged with an external source and then the vehicle's internal electronics will wake-up and turn on the ICE; then the vehicle's inverter will recharge ...slowly..... the small 12VDC battery.
Depending on the age of that particular battery and if it's above 5 years, it would be better to replace it for a new one. My option was an Optima Yellow Top.
Or you can put a trickle- charger for several weeks and see if it will rejuvenate to it's maximum potential.

 

When you stop the car, both the positive and negative leads of the HV battery are isolated from the electrics. (First responders in a crash deserve to know the car is safe to work on victims)
The 12 volt battery is on its own to start the computers, so they can self test and then pull in relays to allow the HV battery to activate the inverter electronics.

Once the HV battery is active again, the inverter will recharge the 12 v battery.

This does not seem to be either of your choices.

Toyota's parts department has a battery with a 84 month prorated warranty.

 

Or the 12V will get recharged when you plugin to recharge your car. @bisco first mentioned this a while back and although I was a little skeptical, I've witnessed the 12V battery voltage go up after it was recharged.

The problem with lead acids is they really dislike being discharged and left that way. You'll end up replacing it in a short while.

 

try testing your 12v before and after plugging in. i believe it is substantially higher afterward.