Suggested approach: watch the clipboard buffer, and whenever it changes (i.e., someone has copied/cut text somewhere), add the new content to a separate temporary list (say, most recent 10 clipboard items), then those can be transferred to the actual clipping files at the user's leisure when they're back out of fullscreen.

A related option, but one I think would prove annoying, is to monitor clipboard, and when it changes, bring up a dialog asking you where to save it (or cancel), but the heavy use of clipboard rather than primary on a touchscreen means you'd be canceling that dialog infuriatingly often.