If you wire it to the brake light, it would come on when the brakes did. If you want it on all the time, it should be as easy as wiring it to any accessory circuit. That shouldn't be a power wire, it should be just a simple switch wire, so anything you can get accessory voltage should work.
I would question why you want to do this though. My guess is this function supercedes (/takes over) any function you would have on your head unit. Meaning if you wire that to full time, then it's going to stay on the camera. I would try connecting it how it should be, and put it in reverse. See if you are able to get it out of camera when the camera is on before you make it permanent.
If it does come out, then what's the point of this? If this is supposed to basically act as a rear mounted dash camera, there are cameras that already do this. I would also want to know if it has a remote or steering wheel controls, (that can function while in camera mode), so you could at least leave it on. I think it would be a giant hassle, but if it stays on the head unit, you could also just wire in a switch to turn it off and on. Wire that trigger wire to a switch next to the head unit to where you can get to it easily. It would always be on camera, but you could turn it off when you need to change something on the head unit. Unless you put it on one radio station and leave it, or USB and just let it go, this is a hassle.
It seems like you're looking for a cheap option, but there's really no way to make the hassle worth it. You can buy what you're wanting this to do, separately.