I have now tried this approach:
procedure TMainForm.ButtonCopyClick(Sender: TObject);
begin
  // Save IEBitmap to MemoryStream as PNG:
  var MemoryStream := TMemoryStream.Create;
  try
    ImageEnView1.IEBitmap.Write(MemoryStream, ioPNG);
    MemoryStream.Position := 0;
    // Load MemoryStream into TPngImage:
    var Png := Vcl.Imaging.pngimage.TPngImage.Create;
    try
      Png.LoadFromStream(MemoryStream);
      // Copy PNG to clipboard:
      Clipboard.Assign(Png);
    finally
      Png.Free;
    end;
  finally
    MemoryStream.Free;
  end;
end;
But it does not work - the image pasted from the clipboard has no transparency:
 