If you don't want to stick something on your screen you could try to find a removable "frame" that would fit over your bike. I have a removable privacy screen cover that I use on my laptop while traveling. That way you could see if you'd have any issues pressing on the screen without committing to it.
Another option depending on your setup would be to cast the bike screen to a TV screen in the room. I've seen people do this with their set up and it can be nice to see the class on a bigger screen and you could possibly put it somewhere where glare wouldn't be as bad - or maybe put a shade over the TV.
I agree this is a pain for a lot of people.