I have configured the browse window copy menu to use different formats in "Copy to Existing Editor" and "Copy to Clipboard" (I don't use "Enclose verse text in quotes" with the clipboard).
When I change between them, the first copy operation uses the format specified for the last copy operation.
For example:
  • Copy a verse to the existing editor - text appears in the editor enclosed in quotes
  • Copy a verse to the clipboard - the text is written to the clipboard with quotes
  • Copy exactly the same verse to the clipboard - the clipboard format is used, and the text is written to the clipboard without quotes

Multiple copies to the same target work just fine, it's only when you change over that there is a problem.