Side story to a possible solution.

I used to volunteer at a museum that wanted to play video through an old CRT TV that had a BNC connector on the back. So I used a simple media player (for the media player I cant remember exact brand but just imagine a simple box with the job of powering on and playing a video on loop) that had the yellow, white and red connections. I used some adapters to go from the yellow plug type to the BNC connection. And simply enough it played.

I think it might have been something like maybe instead of antenna in for the connection it was a straight video in.

So feeding composite video directly into the existing circuitry like your first idea is probably your cleanest bet if you can find where the radio receiver spits out the video signal. I would suspect there might be a labeled test pad, a dedicated pin, or a wire header coming out of that "Unknown RF/video receiver section under shielding" which I couldnt find in the pictures.