I'm not sure if I understand you correctly, but maybe this will help you.
If you're on CPT archive, then there is no "Edit" link, because it's archive and not a page... Archive is list of posts, so there is no single post to edit... Another example of archive page is when you're viewing search results or all posts from given year - there is nothing to edit in these views.
What you can do/how would I do it?
You can add new page and set its slug to the same value as CPT slug...
Let's say your CPT is 'Book' with slug 'book'. You can add normal page called 'Book' with slug 'book'.
CPT archive and single CPT has higher priority than single page in Rewrite Rules, so this page will be ignored by WP. But you can manually display it's content in