I'm going to post a quick reply for closure.
I filed a bug report on this issue: github.com/Scirra/Construct-bugs/issues/7296
Ashley closed it with this comment: "The underlying Clipboard API works in Chrome, but a known issue is currently preventing it from working in the Android WebView. I would guess when viewing in the Facebook app it is using the Android WebView instead of the real Chrome browser so is running in to that bug. I'm afraid there's nothing we can do until Google fix the issue."
So the problem only seems to happen when someone accesses my game from the Facebook link. There are two player workarounds:
1. Type the URL for the game directly into the browser
2. After clicking the link, one can click the "..." at the top right corner of the Facebook browser and select "open in external browser". Then the clipboard will work.