The HV battery "likes" to be in the 40-42% battery SOC range.
If you are not driving the car, and you are just parked, the ICE will cycle to keep this narrow band of charge. That is not very much wattage folks, so yes, the ICE would run nearly the whole time if you tried to pull 1000 watts ( or more ) continuously. However, if you are camping, etc. And only need a couple of 100 watts at a time, or 1000 watts for 20 minutes, you could manually charge the HV battery to 53% ( via leaving the A/C or defrost on in Park, and then turning the engine off manually with the key, and then putting the key back to run without a start. This would allow you to use the battery from 53% down to about 35% on one cycle without the engine on at all. If the HV battery dips below 35% it cuts out to protect itself, and also to save enough juice to restart the car, but does not cut off the 12v battery, so you could drain that one till dead. This would give you about 350 watt hours, or 1000 watts for 20 minutes, per each ~5 minutes of engine run time.

I have been able to pull 1400 watts off the 12v (14v) system without issue for like 20-25 minutes once with engine off.

The 12v battery does not provide power to anything when the key is in run, only when the key is off or to ACC. The HV battery is connected to the 12v battery via DC/DC downconverter, and when the key is in RUN, you can't really distinguish one from the other. The HV battery provides all the juice when the key is in run, if you put a wire tap right on the 12v battery, or cigarette lighter, it doesn't matter.