The pop-up form that is triggered by the widget does not currently include the same functionality that the add idea page in the portal does whereby it searches for similar ideas that might already exist. This is essential to minimize duplicate Ideas.
Could the widget not simply open the Add Idea page within the portal in a new tab rather than trigger the pop-up. Perhaps the simplest solution is to add a toggle that lets admins determine whether they want the widget to open the pop-up or to open the Add Idea portal page.