The camera draws 85mA. I have a 125mA, 10H inductor coming in the mail. The idea is that when the current gets sucked away by the 4A load, the inductor will be there to maintain current to the camera and hopefully prevent the momentary loss of video. If that doesn't work, I guess I can try a much larger 12v capacitor
This is an image of how an inductor can smooth out current draw.
It seems that the camera can run on a range of voltages but it just doesn't like sudden changes in voltage.