If you use the right click and "Copy" something within a Notes document, then it depends on the state, the document is in what happens. If the document is in read- mode then usually the document itself is copied into clipboard. Only Lotus Notes can do anything with this clipboard and other applications will not allow you to paste it.
If the document is in edit- mode, then the selected content will be copied into the clipboard. This can be the attachment, but again only for pasting in Lotus Notes. For other applications there will be only the attachment name in the clipboard.
The solution is simple: Just tell your users to NOT use "Copy", but "Save" or "Save all" from the same context menu. That's the way how Lotus Notes works...
If you need to have a "Copy" action there, then you could make your own context menu, and "mimic" what happens... But I would not do that.