As far as I know, you can not read any app data without running a script. As far as an expression, you could use generate a random number in the range of background you have, and you can seed that random with your composition.
If you are making new compositions with each project with unique names, you would generate new random. If those backgrounds were still images or video files, I would toggle their opacity if their number was called. If you want to replace the background layer in your project, you can also write a script to do that every time you run the script.
bgcount=10; \\assuming you had 10 backgrounds
seedRandom(thisComp.name, timeless = true);