Actually you are quite wrong. The Jockey Club which is the organization that registers Thoroughbreds, does not allow artificial insemination; all offspring must be the result of live cover. Since the horse must be registered to participate in Thoroughbred racing, any offspring of must be the result of live cover.
Based on the way a horse's semen is collected in which it must stand on hits hind legs, there is no way they would have gotten any from Barbaro during his long recouperation period.
So the hundreds of thousands, perhaps even a million or more in vet bills they spent on Barbaro is down the tubes as far as the owners making any kind of profit is concerned.
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.