I assume you don't need audio switched so only video. My suggestion is to get a video MUX unit. They will route a video signal from multiple sources to multiple outputs. For example, I have a 4x4 MUX that will feed 4 signals to any or all of the 4 outputs. They can be pricey depends on the level, from like $150 to $1000 but you may be able to find a good solution for about $200. One thing, mine has S-Video input/output. I am not sure how easy it is to find RCA MUX but I am sure they exist.
One
example of it is here. It's 4x2 but only for S-video and composite video.
Again, do some search on Google. I only did a quick check but I am sure you will find what you are looking for. Good luck.