Tried an experiment, span up a toy gyro with 3/16 shaft using string. On the tacho this got to 4500rpm but dropped fairly quick to 3000. I reckon you would be lucky to do much better. Using the criterion of a gyro that will sit horizontal and do 1 rev/sec with a rotor speed of 2000rpm the 75mm diameter rotor is marginal – OK if turning a bit faster but losing energy fairly quick.
The snag is that a toy gyro to sit horizontal on a little tower is pretty challenging, the weight (mass) cannot be too much but you need the mass to get the gyro couple – and the practical rpm achievable is in the 2000 to 3000 region. This design criterion is quite different from designing a gyro compass.
As a kid I remember a pretty good gyro that was about 70mm diameter with a thin ish rim – but made of lead painted in shiny brass lacquer – probably not allowed now….