This is the code that works, cooked down:

    dsFahrauftrag.Params.Values['Fahrauftrag_Nummer'] := IntToStr(FahrauftragNummerAsInt);
    dsFahrauftrag.Open;
    try
      dsFahrauftrag.LoadRows;
      if dsFahrauftrag.RowCount = 0 then begin
        Application.Request.SendError(HTTP_NOT_FOUND, Format('Unknown Entity "%s"', [FahrauftragNummer]));
      end
      else begin
        Application.Request.SendContent('Got it');
      end;
    finally
      dsFahrauftrag.Close;
    end;
  except
    on E2: Exception do begin
      Application.Request.SendError(HTTP_INTERNAL_ERROR, E2.StackTrace + ': ' + E2.Message);
    end;
  end;

Nothing fancy. Now, put a "exit;" right after the first 
"Application.Request.SendError":

...
      if dsFahrauftrag.RowCount = 0 then begin
        Application.Request.SendError(HTTP_NOT_FOUND, Format('Unknown Entity "%s"', [FahrauftragNummer]));
        exit;
      end
...

Then, you get a second resonse from the E2 part, saying

Error 500: Exception instance raised in unit WebData at 2517,4
TDataSet.Close in unit Main at 109,21: Value conversion error

Something goes totally wrong with an exit; in this try/except block, killing the 
.Close method.