thanks a lot for this hint which is really helpful. So far, I have also been using a keyer with a built-in tone generator (Ultra PicoKeyer by N0XAS) to transmit CW audio in SSB mode. With a good and narrow TX filter, this worked quite well in SDR Console.
The solution you suggested works much better for me. However, I had a hard time getting it to work. I soldered the cable for the serial port (luckily, my laptop still has a native COM1 port), and it worked at first try with a hand key. But that was not what I wanted. So I started playing around with different keyers I had in the shack. The PicoKeyer would not key the COM port at all, my K1EL WinKeyer sort of worked, but not very reliably. So the serial connection seems to be somewhat sensitive. Then I tried my Palm Radio Code Cube (which they unfortunately do not build any longer). This one worked right out of the box. I suppose the other keyers require an additional driver stage or even a relay.
The combination of Morse Keyer 4.3 and SDR Console really produces a very nice and clean CW signal, even without additional filtering!
73 and CUL in CW!