There are also Write/Web Save permissions given to the user on the workbook.
It doesn't work with Write/Web Save? Have you also enabled Download/Web Save As, or is it only Web Edit and Write/Web Save?
Download/Web Save As is one that you also need in this case.
I am having the same issue. I do not want to give users the ability to Delete, so I have created custom permissions. However, users can only ever Save or Save As when I give them full Editor permissions. This makes no sense to me, but unfortunately is how it is.
The permissions in the image DO NOT allow users to Save or Saves As for some reason.
I think this will help you: Grant Edit and Save Permissions
So I would first set the permissions at the project level. But do not assign permissions to contents. Then set permissions on the individual workbook as seen above.
Let me know if this helps!