The car has to be "running" for the DC-DC converter to maintain the 12v battery. Since you have a 12v battery, I'm assuming it's a PHEV. That battery is very small, and has little capacity of its own. It's really there simply to power the basic circuits of the car, while the traction battery keeps everything going. It's simply not capable of supporting a long term drain with an inverter connected. The car has to be on to keep the battery charged. Also, the DC-DC converter in the car is only rated for about 1900 watts. And that rating likely isn't continuous, so it can't keep a 2kW inverter running at load.Did you end up connecting your inverter to the nuts on the battery posts? I was able to connect a 2000w inverter running a 700w fridge but the 12v battery drained quickly and the inverter shutoff after a few minutes. I had the car in utility mode but am wondering if I hooked it up wrong. I would like to use the inverter for camping.