If it's a local thing, where the normal latency of an internet stream would be too much, you might add a display output from your OBS machine, and then right-click OBS's Program window -> Projectors... Send a full-screen projector to that new display.
The projector is silent, so you'll have to use OBS's Monitor to get the audio out, and then send that Monitor to the same display output if it's something like HDMI that supports audio too, or to another device that you send to the Nursery alongside the silent video.
Or, you could use another output from your audio console, to package together with OBS's silent video. That can work too, and would preserve your headphones in OBS if you value them that much.
