supposedly you CAN breed your homebreds....

go to training-stables-trainer-horse on the right of their action icons should be one for "export" make sure you save the system file once exported. The horse will save at it's peak stats and according to the manual this will allow you to breed with the homebred. You can only export once it's made it's first start.

I did this with my first (a white!), but I didn't see her as an option at the following breeding season -- I think breeders have a randomizer in play like there is in delegating to the assistant trainer.