How do I make ClickToCopy copy some text rather than string?

You can use CopyToClipboard:

Button["display text", CopyToClipboard["file name"], Appearance -> Frameless]

CopyToClipboard >> Scope

"Copying a string copies the contents of the string."

Column @ MapThread[Button[#, CopyToClipboard @ #2, Appearance -> Frameless] &, 
    {{"display text 1", "display text 2"}, {"file name", "Alex E"}}]

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