That story comes from Mosolov's memoirs,
At the Court of the Last Tsar. You can read them in full
here. (The part about the sable is in Chapter 1, section 3, "
The Children.")
Short version:
The sable was tame, and the couple had been checked out by the palace guards. Alexandra was in the room with the children while OTMAA played, and the tsar was informed of the whole thing before he went up to the playroom.