Without the Steam Controller, I didn’t want even play docked on tv / monitor. The only time I tried it was to record some gameplay (an external recording device in between monitor and the Deck), so that I get everything record, not just the gameplay. I don’t have the original dock and tried with some random HDMI output thingy. And it worked! Here some bad illustration of the setup:

From Steam Deck over converter, to recorder. The recorder splits up to tv output and saves the live video recording to drive.

[ Steam DECK ]  o--o  [ USB to HDMI ]
                            o
                            |                 -----------
                            o                |           |
  [ USB Drive ]  o--o  [ Recorder ]  o---o   |    TV     |
                                             |           |
                                              -----------

And it worked perfectly fine! Did I need to do the ascii illustration? No, but I had fun with it. :D