The Red/White wire (Pin 1) corresponds with the K-bike''s unswitched 12V line and Black (Pin 4) corresponds with the K-bike's switched 12V power line. You need both present and working properly on your Tenet. These both are referred in the instructions for your gauge as +Anode (weird). Power return pins are labelled -Cathode (weird again) in the instruction guide correspond to K-bike ground. Use a voltmeter to verify that you have power on output Pin 1 with the ignition off. Cycle the ignition switch to verify that you always have unswitched power available at all times and in all ignition switch positions. Measure it relative to Pin 4 and not the battery negative terminal or K-bike ground.
To put the gauge in program mode, it should be enough to switch off the ignition, press and hold the button on the back of the gauge and turn on the ignition. No need to remove the Red/White wire. Try that.
You're right - there is no NVRAM setting on the gauge corresponding to MPH/KPH. I still believe that setting should be held by the unstitched power and should survive ignition switching. If you loose unswitched power as you cycle the ignition switch, then definitely your KPH/MPH setting will be lost.
I've attached the user guide I wrote for the cheap Ebay gauge to this post.