To see what will happen in the case that you describe think of that same setup, with one of the gyroscopes removed.
If you release the gyroscope then it will settle in steady precessing motion, gradually sagging down because of the non-zero friction at the central pivot. (I bought the particular model of gyroscope that you see in the video, I very much enjoyed playing around with it.)
You can replenish the energy lost to friction by giving the precessing gyroscope a little push in the precessing direction.
Now the bit that answers your question: if you push a bit more, making the precessing gyroscope go around a bit faster than the natural precessing motion, then the gyroscope will climb.
The friction at the central pivot act as a mild counterforce, preventing the gyroscope from precessing at the natural precessing speed, causing the gyroscpe to sag. Conversely, if you push in the direction of precessing motion, then you make the gyroscope rise.
In the two-gyroscopes setup the other one will have the same response to pushing to faster-than-natural-precessing: it will tend to rise.
Given that the two gyroscopes are on a rigid rod the tendency to rise will transfer to the center. So yeah: a scale will show a lower weight.
Of course, this does not provide a way to get lift for free: in order to push you need leverage; your feet need to be firmly on the ground. (Compare pushing a large piece of furniture: if your feet don't have grip then you will only push yourself backward instead of moving that piece of furniture.)