How to share a live video feed between computers?
 

This question pertains to our (very!) small church. We have one computer, in the sound booth, which is connected to a projector and handles PowerPoint presentations with song lyrics and occasionally a sermon outline. We also use it to play the occasional video of ongoing missions ministries and similar.

Right next to it (and sharing a monitor and keyboard with a KVM switch) is another computer (both machines are Windows 10) running OBS Encode to take a video stream from our installed security camera and transcode the rtsp stream from the camera controller to the rtmp stream format required by our streaming service. We have good LAN bandwidth capability, a combination of wireless and hard-wired (Cat 5e or better).

What I would like to do is to be able to take the second monitor (projector)'s output from the first computer and have it available in the second computer as a video source which OBS Encode can see and which we can switch to during the service if desired. And, perhaps some time in future, I'd like to be able to add capability to the first (PowerPoint) computer to take a display feed from, say, a computer at the pulpit and output it directly to the second monitor (and from there to the encoding computer as well).

What are my choices for implementing this idea?

So ... huh? :question:

You have two computers.
- A/V computer
- surveillance

Both are KVM'd.

A/V computer has 2nd output? HDMI? DVI?
And it's connected to a projector?

Okay.

The camera bone's connected to the NAS bone. (Surveillance Controller in a Synology RackStation)
The NAS bone's connected to the Encoder bone. (OBS encoder running on a Windows 10 PC)

What I'm wanting to do is to take the HDMI projector output (monitor 2) from the PowerPoint computer and to pipe a copy of it into the encoder computer as a video source which OBS encode can 'see' and which can be (manually) switched to during the streaming/recording of a service. The projector output needs to remain unaffected, of course. Something along the lines of an HDMI splitter and capture device might work.

Related, but separate, is the question of whether it's possible to connect the projector output from the PowerPoint computer to a laptop at the pulpit, in effect using it as a monitor-over-IP. I haven't even really looked into that possibility, but it strikes me that a solution might be out there.