*- Unless I've missed something obvious -*
There doesn't appear to be an option to add an attachment from your Briefcase.
I've searched all of Zimbra and can't find it mentioned, so must be missing.
When I go to add attachment, there is a tab "My Computer", seems logical that a Briefcase tab here would be the solution.
Seems a bit pointless to download a file from the Briefcase, to only upload it again.