Do task stack work on handler to ensure in-order execution

The rest of the ClientMonitor work is done on a handler to guarantee
in-order execution. e.g. setting to null when auth completes, updating
mCurrentClient when new client is requested, etc. Put this work on the
same handler.

Fixes: 137237140

Test: BiometricPromptDemo, go home while authenticating multiple times;
      no crash observed

Change-Id: I43126ce13fb88d751571d0d52fe594c0293f794c