So is it allowable if it's artificially insemenated offspring has offspring under live cover? If so, his line can just skip a generation of racing, but will still make some money.
No.
No. All breeding of Thoroughbreds must be by live cover. No artificial insemination allowed anywhere in the pedigree to get a registered TB.
Semen from Barbaro would be worth nothing in the Thorobred world. I do not know about other breeds using him for cross breeding. But trying to create a racing Thorobred offspring from Barbaro would not be registered by the Jockey Club nor allowed to race.
In his condition, Barbaro would most likely not been able to mount the dummy mare for collection, anyway. There is a method which involves a person holding the collection sleeve, but in his delicate condition with his balance, etc, I would not think they would e3ven have tried it.
No offspring not of live covere will be registered as a TB.