I get the correct signal on the TX pin (using the UNOs TTL2RS232 converter it's even showing the correct output in the IDE's serial monitor).
This makes no sense. Either you still have a chip in the Uno board (which you cannot have when using it as a serial converter only) or you haven't shown a correct schematic when testing as there is no physical connection shown between the at328 on the breadboard and the USB connection.