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.
So to clarify what specific behavior you are requesting to be implemented:
If a user has opened a vault item for editing, then clicking the pop-out button should launch the pop-out widget with the starting page being the edit window for the item that was open in the browser extension? What if the user had already started to make some edits in the browser extension before clicking the pop-out button — is it OK if those edits are lost, as long as the pop-out still shows the (now cleared) edit window?
What about something like a vault search? Here, the pop-out button is actually available already in the current implementation, but if the user has entered a (possibly complex) search query and started scrolling through the search results, then clicking the pop-out button will clear the search. Is this behavior consistent with the request you are making in this thread, or do you also wish for the state of the browser extension to be transferred to the pop-out window when it is launched?