This is a good topic, and not just for teleprompting or for people with sensitive eyes.
I like to sit in the subject’s chair and look toward the camera, to see how I can make the setting more comfortable for them. 2 tips for that:
(1) Use a light to brighten the room BEHIND the camera, so that the key light doesn’t seem so bright.
(2) Put a big scrim in front of the key light to make it more diffuse and less harsh on the eyes.
As for the “gray screen,” that’s a mystery. Could the teleprompter monitor be too bright or the text too contrasty? Would a different color help? I’d recommend that you just ask the other studio, if that is possible.