I wouldn’t be so quick to call the breeder “irresponsible”. Some breeders fall on hard financial times. It can and does happen to anyone. Especially true when the financial crash hit starting in 2007. I would bet there are a lot of unregistered TBs from those years because of unpaid stud fees. I am sure there are lots of them in any given years.
Who ever has the mare not is getting some money out of her. I would bet the breeder had a LOT into her at one time and never saw a dime.
As to your question. The JC will not verify anything other then there is no Stallion Certificate on file. This has to be released by the stallion owner/s in in paper form or electronically directly to the JC before a horse can be fully registered and the foal certificate issued. NO IFs Ands or Buts. The stallion owner technically has a lien on the the horse.
It is quite expensive just to register a horse that is over 2 years old, $2,025.
If you are prepared to to pay that then you will need to contact the farm where the stallion stands and ask for mercy. Try and and work something out for a much reduced fee. Or no fee. If the horse was a gelding you would have a much better chance if he is past an age worth racing him and the fact he has no breeding value.
A mare is a different story. The dam of California had little value until he came along. He had 3 half sisters that were basically given away. One of which only months before he hit the track. 6 months later the lucky owners hit the lottery. I am sure all the calls and money offered shocked the crap out of them.
You maybe able to work something out with the stallion owner by offering to have her pedigree stricken/removed from the stud book right after registering. Therefore nothing out of her could be registered as a TB.
Not sure how this would effect her eligibility for other breed registry.
If you want to give the stallions name I can give you a pretty good idea if it is worth trying to get something worked out. Assuming it is a “known” stallion not something standing in the hinterland in someone’s back yard.
If that is the case you should have a much better chance of getting the stallion certificate for little to no money. Some money is better than no money to these types of owners.
Good luck!