Don't know how accurate the info below is, but might help narrow the search.
Am wondering if this horse ever was registered with The Jockey Club as "King Charles." Maybe not. Could've been renamed when put into show horse training. Also, movie trivia sources don't say that he ever raced in real life.
Hope someone can figure this out! :>
From the web:
http://horsefame.tripod.com/1Nvelvet.html
His real name was King Charles a seven year old thoroughbred - a red sorrel with a white stripe on his face and white socks in all for feet. The Rivera Country Club in West Los Angeles, California was where King Charles and Elizabeth first met. This horse was owned by a society woman and trained as a show hunter and jumper. The owner offered King Charles for Elizabeth to ride and she grew fond of the horse and eventually became the equine star of the National Velvet movie.
