Funny - just a few minutes ago, before reading your message, I learned while installing https://github.com/stig/ox-jira.el, about =org-export-copy-to-kill-ring=:

org-export-copy-to-kill-ring is a variable defined in ox.el.

Non-nil means pushing export output to the kill ring.

This variable is ignored during asynchronous export.

In my brief testing, it does exactly what you want: regardless of the export output or method (to a file or to a temporary buffer), the result is placed in the clipboard.