Legally, if under the divorce the dog became solely yours, then she is yours and the AKC *should* accept that. But if they won't, the only way to force them to do so would be to sue them in county court (not small claims; "regular" court) and get a court order (a "declaratory jugment") that the dog is yours and they should register her that way. Only a court order can force people to do things they are unwilling to voluntarily do.