Slightly off-topic, but I've been wanting an HDMI Freeze button for a while.
I looked at the auvidea link that kayfox suggested, and could probably implement it in software on an RPi. But that has limited resolution and frame rate (25 fps).
For a feature built into most projectors, I'm surprised there's no products to do this.
The resolution and framerate issues with that solution appear to be software implementation limitations, I haven't looked into them, but I would expect they could be resolved, since they appear to be an artifact of just one developers environment being tested against.
Mouser: https://www.mouser.com/ProductDetail/Auvidea/70501?qs=EU6FO9...
Datasheet: https://www.mouser.com/datasheet/2/864/B10x_technical_refere...
Edit: Theres also this one on Amazon: https://www.amazon.com/gp/product/B0899L6ZXZ/