I think you would be better served using gears (which you can get real cheap from most hobby stores) so that you don't have to worry about slip.

 

I happened across copper tape and saw that it is backed with "conductive adhesive". A quick jump to Google reveals many, many hits under that term, including this from 3M. I'll bet 3M might offer advice for your specific needs.

 

I like PaulKTreg's idea of the opto-sensor - sensing a black or white line on a white or black surface.

You can put the sensor near the roll and mark the back of the screen when it is near fully extended. That way, there will be only minimial sag/sway.

This is essentially the line-following robot that a 15-year-old builds in high-school robotics club. Very easy. Make the line wide and set gain high. You should be able to sense at 5 to 10 inches if you need to.

 

But he's already concerned about tape on the screen being too thick and stiff and distorting the screen, either when it's extended or when it's rolled up.

The magnets I have are a common size, 3mm diameter and 2mm thick. That's tiny. I'm sure he could glue one of them behind the screen or on some moving part without affecting the screen's ability to fold up?

 

I exhausted all the hobbyist-level black/white sensors I could find, and everything I could find had a sensing range <.25"

I started looking into industrial types, and surprisingly I found these for ~$10/ea. They're made for paper industry, with a range out to 12." I think I might go ahead and order a couple so they're waiting for me when I get home from vacation.