What does she mean, "not allowed"? She's not the boss of your daughter. Couldn't you just tell the principal she can't forbid your daughter from riding a bicycle?
I would try to record this conversation, and once I had the recording, warn the principal in a public letter that would be a criminal act and cause for a civil suit, and threaten to also take it to the local news media. Of course, if my daughter was willing to take up this fight.