Installing dashcam on auto-dimming rear view mirror

I have a 2012 Prius Five with the technology package, and it includes an auto-dimming rear view mirror. With the cost of good dashcams dropping, I was thinking about getting one and mounting it to the backside of the mirror (facing forward). I was hoping to be able to tap the power that's used for the auto-dimming mirror, rather than having to run new wiring back to the fusebox. Has anyone tried this, and did it work?

Related issue: my mirror slowly sags. Once every three days, I need to tilt it back up. Any known fix?