In addition to the suspension, which will take effect from Tuesday, March 9, Elliott was also ordered to pay costs of €15,000 (£12,921.45).
The Queen has been without corgis at her side for more than two years after her final one, Whisper, died in October 2018.
James, who rode Milan Native to victory at the Cheltenham Festival last year, said he is 'heartbroken by the damage' caused by the video.