I made a quick, and, somehow, dirty, fix. I use “browser.windows.create” to open a new window in the browser once the “import settings” button is hit.
Inside this window is the file chooser and the “overwrite settings” confirm button. There, the user can freely select a file and import the settings.
I am aware, that this is not pretty but I do not see another way of keeping the addon popup open once the file chooser is shown.