My car doesn't turn off my lighter port when car is turned off and I'm not sure I'll remember to unplug it every time I get out of my car which could result in a dead battery. :/
There are dash cameras that are designed to be run 24/7, and you typically need to hardwire them to your battery. That way you can capture any vandalism/comprehensive damage that happens when the car is parked. From what I understand, they can go several days recording on the cars battery, so you could check the actual power usage or look for an always-on camera system that is designed for such use. You'd still need to run the car every couple days to keep the battery fully charged, but if you drive a lot it would work fine.