I don't have experience with betadine, I've tried neosporin abd goldenseal with success. Also administer reptaid
http://www.reptaid.com/ as per the instructions and make sure the temperatures on the high side of the proper range. If there's not a change an amputation done at the vet would insure the infection doesn't progress into a serious problem. Apple cider can be applied topically and mixed in the drinking water occasionally to strengthen the immune system and fight infections. If you do get the tail amputated it would still be a good idea to treat this way to ward of what may have caused this.