spot light vs go pro camera.

I think you'd be better off getting a camera that's intended for constant power and is actually designed to be hooked up to a monitor. The GoPro (I have the Hero 3 Black and love it) is designed to be a stand alone unit that records on it's SD card/Internal memory. You can monitor wirelessly through an app, but I don't think it can be wired to a monitor easily. And unless they've changed something between the Hero 3 and the Hero 4, it's not setup to have constant power running to it like you would need in your potential application.

I would guess that the spotlight and it's wiring could be left as is and the camera mounted above the light. The wiring for the GoPro camera will have to include power for the camera as well as a data cable to view the images. The 12 volt power can be run to the camera with a mini USB connection and another micro HDMI cable would have to be run to transfer the video to your monitor.
The biggest problem I see is that the Go Pro is not waterproof. They do sell water proof cases for the Go Pro but these are designed to be used underwater and do not have provisions for the two external cables that yo would need.
I love the GoPro camera and use on on my dash but I would think that a home surveillance camera would be better for what you need. Most of these are run on 12 volts and they use a single cable to supply power and video transmission to a monitor. The external surveillance cameras can be found that have 720 or 1080 resolution and are already designed to keep out the elements. Attached to your light they should be able to view the surrounding area as well as the GoPro. Some of the surveillance cameras can be hooked up to their own joy stick and can be rotated, tilted, and zoomed(like they do in the Vegas casinos) to provide even more coverage than you can get from a GoPro. It just depends on how much you want to spend and what resolution you want.

I also have a go pro 4, and it not going to be a easy fit. To fit power you must take out the side button and wire in a 3.8 volts, not 12volts. It also only streams and memory chip output. I would go to a back-up type made for the front for constant on. My new car has front and back cameras . I can select either as need. It records to a 64gb memory card. Check them out at any good backup cameras system store on-line

Thanks for the info on the GoPro, I think a camera that is made for the application is a better choice.

Go Pro cameras are fantastic. I have two. I use one for filming with a drone and they are very flexible. But.....for what your wanting to do, I'd suggest something made for that task. Yes you could Micky Mouse a GoPro to work but there are other camera that don't have to be Mickied.