The Tale of the Whispering Walls - S2-E7
Factual error: Betty Ann is supposed to be telling this story on a Leap Year night with a full moon. While it's necessary for the plot, the most recent Leap Year with a full moon was in 1972, and is calculated to happen in 2048 next.
Factual error: Batman is in Japan and tracks down Kyodai Ken by using Caller ID, which returns a seven-digit "555" number. Telephone numbers in Japan have 8 digits.