[fy20]

> Not to mention the main battery would have to constantly recharge the 12v battery, which is probably bad for both batteries.

I don't know about a Tesla, but this is how regular ICE cars work. The power from the alternator varies depending on the speed of the engine, so if you are stopped at a red light with your radio, wipers and head lights on, the alternator may not provide enough power. This is why the 12v battery is still connected after the engine has started, not just to charge itself, but effectively as a big capacitor to balance the 12V power supply to the car. As long as the inverter is normally drawing less power than the high voltage to 12V system can supply, it shouldn't damage the 12V battery.

This is also why cars with start-stop usually use AGM batteries, as the battery will be run down a lot more while the car is stopped (and the engine isn't running), which would quickly damage a regular lead acid battery.