This would need additional changes beyond just adding the button. Currently, the widget GUI is refreshed on pop-out (or more likely, a completely new widget process is started in the pop-out window). There would still need to be code to capture the unsaved state of the browser extension and transfer this information to the pop-out widget.
So, while making the pop-out button available elsewhere in the GUI is not a bad idea, I don’t think it will solve the main problem.