To answer your question simply, not directly in the ui. What you are asking about it called using a Render Texture. Many game engines have this option that can do it directly without compositing. Poser doesn't know how to do this directly in the ui and using the camera render output as a material within the material room, which means it has to do it the compositing route.

This all has to be set manually in Poser, or scripted. Basically the scene will have to be rendered for all of the additional cameras that power screens in the scene, then the intended images placed on the screens within the scene for the final render. Each of the cameras will have to render out the scene by itself first, saving the renders sequentially in their own folder(s). Then apply the texture(s) to the screens using a movie node with the sequentially saved renders to do the final rendering. 

This could be done with scripts, but it is still going to render out everything and all the script will do is automate putting it together for the final renders. Everything would have to be internally named properly, saved to individual folders and sequentially, etc...

I don't recall anyone ever asking for this in Poser, and that doesn't mean that you could not ask for the option either. It would still have to render out everything unless the cameras and screens were all along the same line. Game engines also do this by rendering out everything separately if more than one camera is used in the scene. They do it one final frame at a time, for every frame that uses it, for obvious reasons thou.

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