Is there an ARINC label/data that can be associated with a gyroscope's precession?
Short answer: No easy way. It may be possible for a crew to detect some drift by comparing the IRS mag heading to the magnetic compass.  This does have problems though as the IRS doesn't really 'measure' mag heading.  It works in true heading and calculates mag heading by applying a magvar correction from a table.
Modern aircraft do not have free gyros that output data. They have inertial reference systems (IRS). The IRS has multiple gyros and accelerometers and outputs a complete navigation solution.  They also have an internal GPS (or an input from an external GPS) to improve their accuracy.  So there's not a lot of 'drift' in the output solution.
These units have internal integrity monitors and will report a fault to the crew if the nav solution exceeds limits.  In some failure modes the IRS will revert to an attitude only mode (ATT) that just outputs raw rates and accelerations (free-gyro mode).  In this case, the crew can provide a heading update to set a heading.  This will drift just like a stand-alone directional gyro.  The crew will have to check it regularly against the mag compass and update as necessary (~ every 10 minutes).