The authority told him he had two hours to change it, he said, or the horse would have been pulled from a race the next day at Beverley in East Yorkshire.
Tasked with providing an alternative, he decided upon Sling Yer Hook - another local phrase meaning "go away".
"It's ludicrous, ridiculous, bureaucracy gone mad," said Mr Rhodes.
In the end, the horse came seventh in the race, after being upset by a long wait in the stalls, according to its owner.
Speaking about his appeal, Mr Rhodes said: "I just hope common sense prevails and he's allowed to keep the name."
A BHA spokesperson said that although Weatherbys - who carry out administrative functions for British racing - had approved the name, "on further consideration it was viewed as inappropriate" because of its potentially offensive reading.
The BHA said it understood there was no malicious intent on the part of the owner, but that many people would have been unaware of the word's meaning.