I'll compare notes with you. When I'm away I use an old 6mm 4MP chinese market dahua camera pointing at my touchscreen thermostst. At this point in time, this is considered a fairly low-end camera. It's very close to 32" away and works great. One thing I noticed is, if I have this right, the horizontal FOV of your camera is 120 degrees. The camera I'm using has a horizontal FOV of 55 degrees, and the thermostat is still pretty small in the image, which leads me to think you're using a really really high level of digital zoom. This would work great if you were in a TV show, in real life, not so great. So my conclusion also is that you need a camera with a narrower FOV, which is in effect, optical zoom.
I'm reluctant to recommend a specific camera because there are so many variables and assumptions involved. My thinking is that with the backlit display you don't need a camera with great low light sensitivity and can go at the low end of the price range. Nothing wrong with a higher-end one of course. Taking a quick look at Andy's cameras I see only 2.8mm and 3.6mm in the lower end fixed focal models. The varifocal T22IR-ZAS-S3 can go to a 30 degree horizontal FOV and has a close focus distance of 31.44 inches. With my assumptions this sounds ideal but you never know for sure until you try it. If I were in your shoes I'd consider getting a older used 6mm or varifocal for somebody on the forum. Wouldn't surprise me if there over a hundred of them taking up members' closet space.
