Unfortunately, I don't think this will be possible without development - either by modifying AtoM directly, or possibly via a custom theme plugin. Manage menus module will allow you to customize existing menus, but adding a whole new icon and drop-down menu will require code changes.
I have dug up the commit that includes the first time the Clipboard was added to AtoM - obviously it has a lot of additional functionality that you might not need (such as showing clickable clipboard icons throughout various AtoM search and browse pages, and including a count of pinned items), but if you are interested in exploring the development option further, it might give you a sense of some of the files that were modified to add it to the header:
You can see where various components are added to the AtoM header bar in this file:
Beyond that, I will have to pull in a developer to assist in responding, so let me know if you are going to explore development further and if so, how we can help!