You need a very well balanced gyroscope for this.
If it is not properly balanced the drift due to the imbalance can be more significant than the drift due to Earth's rotation or change in position.
So I would recommend one which can be calibrated.
Set it up in one location, leave it fixed, monitoring how much it drifts and calibrate it so it doesn't drift. Then carefully transport it to another location and see if it drifts there.