This is all just theory, but based on @GSSRyan's JSS Flask and my PHP knock-off, you could accomplish something with a scriptable web page and web clip.
Make a extension attribute for "eBackpack Status" and a smart group where the criteria IS "broken". Scope this group to the exclusions for eBackpack. You'll then distribute a Web Clip to all devices called "Fix eBackpack" to a webpage that can pass the ID of the device into the URL as a variable. This web-script would write "broken" to the "eBackpack Status" extension attribute, and update the device's inventory—I'm fairly certain an update is needed for the JSS to recalculate smart groups when dealing with extension attributes. Now your troubled devices should be in the exclusions. Now to get them out...
Getting a bit crazier (and even more theoretical) you might be able to set some sort of timer on the page, so after 60 seconds another api request is sent (or the user is told to wait and to push a second button) to revert the extension attribute back to normal, thus removing the device from the app's exclusions and triggering a re-install.
Edit: Having re-read your post, using JSS Flask would pretty much be what you need. Students delete the app, open the update inventory Web Clip, device updates, JSS sees the app is missing, puts it back in self service.
And as of JSS 9.93 there's a "Display app in Self Service after it is installed" check box on the app records now—that may be the simplest solution. It'll keep the app in self service, even if the JSS thinks it's installed.
