Adobe Edge replaces the project html file on every save.
How do I add custom HTML (e.g. a script tag) to that html file?
You can do it.
Open the project in Adobe Edge and save all your changes.
Edit the HTML file with an editor of your choice while your project is still open in Adobe Edge.
Save the file in the editor. Switch back to Edge. You will be promped to reload the project like this:
"index.html was changed outside of Edge Animate. Do you want to reload the composition? You will loose any unsaved changes."
Click YES.
From this point on your changes will persist whenever you save the project.