Back in Stock App is designed to work only on published (live) themes. At the moment, we don't support unpublished themes as we don't have access to the code files of unpublished themes.
Once you have published your new theme, the button should be properly displayed on your product pages when the product is out of stock..
If that’s not the case or if you want to make any changes to it, you can contact us and we will make sure it is working properly.