I have never dealt with APHA, but AQHA will re-issue Registration papers with a court order if the courts rule that she is yours and you should have the papers. They may require DNA testing to prove that she is who she is supposed to be. The only times I have had to deal with this was with AQHA and at that time I only had to supply four photos of the horse and a copy of the court order. I just got the court order through small claims court and my total expenses (except for the board bills and training bills against the horse) was less than $100.00.
Added. I would first write APHA and tell them that the 'owner of record' will not give you the papers. Send them a copy of your Certified Check. Tell them that they will not give you the original papers and a transfer. They will ask their side of the story and will demand the papers from them or threaten them with sanctions and/or they will be denied APHA privileges.
Added some more. This IS NOT an 'illegal sale'. Don't call it that. No law was broken. It is strictly a Civil Matter. Possession and the most recent feed bill should over-ride everything else. You just need to be the aggressor.