I've spent more than hour, but finally configured it. So I worked on CMS Editor role, which had default security settings:
Design tab is not allowed - which what you need
Properties -> Templates gives 403 error
So I went to CMS Editor role edit screen -> Permissions:
Permissions for: Module
and checked Allow for Design Website permission. This enabled Properties -> Templates area for the editor, but design tab remains invisible (it is denied in UI Personalization section).
So looks like this is solution, but it might potentially allowed access to some other sections for CMS Editor, but I'm going to leave this for you.
Hope this helps.