As I understand the sequence in how the 12v battery recharges,,,
When it gets to a certain low level, it engages the relays to the main traction battery, 480v? and it is then recharged. If the 12v battery becomes low and is 'cold soaked', it can drop too low to be able to activate the relays for the traction battery, it then will fail to be able to recharge and it goes into a failure status.
This same fault reasoning happens when the 12v battery has exhausted its ability to hold a longer charge, due to too many charge / discharge cycles.
Keep in mind that this 12v battery is what supplies all 12v devices, lights, radio, fans, dash, heating / cooling ect. The main traction battery is it's only source for recharging. And if the 12v battery can't pull in the traction battery relays. it can't recharge.
It has been suggested that Tesla should use a 12v lithium battery but I do believe that the 12v lead acid batteries that Tesla uses, have a better 'Deep Cycle' fault rate. Meaning, you can draw the battery down real low and fill it up many times before it will fail. Lithium batteries, not so much. That's why the Main Traction battery will always keep a reserve amount. Drawing down that reserve will "Brick" the traction battery.
As to using a 12v cigarette charger, the car turns off the 12v outlet after a period of time, so it would not serve to back charge, Coleman and others offer an OBD plug setup to back feed solar and or 12v chargers but personally, I wouldn't risk tying into the cars circuitry in this manner.
So this brings us to the terminals for the current on board 12v battery.
There use to be terminals under the front plastic nose cone, but I think those have been moved?? and therefore are no longer available?
Not sure how the Tesla Techs jump start the car now.
Does the Tesla have a 12V cigarette lighter outlet? If so, I'm sure you could use a 3rd party battery tester on it? Or perhaps pull some wires iinto the frunk for this, and check it whenever you feel like it, and al least every time you top up the washer fluid.