@enpointe19 She can not legally take your puppy to breed. She shouldn't even breed a 9 month old puppy. They need to be at least 2 years old. That is ridiculous. Ownership of the dog (unless dictated by a contract) can usually be determined by vet bills and/or microchip. If you pay for the vet bills and food to take care of this dog, then you have a stronger case that this dog is yours.
AKC does not dictate breeding rights exactly. There is full registration, which means if you decide to breed this dog with a dog of the same breed, you can register her litter with the AKC. If the breeder were to give you limited registration, you can technically still breed the dog, but no litters can be registered with the AKC. AKC has no control and actually doesn't want any control with the business practices itself. Just because she registered the dog doesn't mean she has breeding rights.
If this dog is legally yours, then it is your decision. She has no control over that. If she relinquished ownership of the dog to you, then she relinquished the right to breed her.
The ONLY time it is possible for a different person to have breeding rights than the owner is whenever you are a part of a Guardian Home program. This is where the breeder gives you the dog but has the right to breed the dog whenever they want. There is a lot more that goes into it, and there are very detailed contracts stating what all is expected in the process and if the owner can actually handle it. Good breeders don't make this option available to just anyone. Anyway, If your ex does not have a contract, then she can't do anything about it.
If she keeps coming at you, I would threaten legal action. Tell her that is the only way she is going to get the dog. Also, if you go and get the dog fixed, she can do nothing about it. If needed, some vets will even say that it was a needed surgery to get the dog out of a bad situation.
I'm not going to tell you not to breed your dog. That is your decision, but please do not breed this dog as a puppy. That will be so hard on her. Babies shouldn't have babies. If you decide to breed in the future, do your research on how to do it right. Research about different health tests to do with the breed. Some will have DNA tests along with hips, elbows, heart, thyroid, eyes, etc. Not always all of those, but each breed is different. Also, always take back any of the puppies. We don't want to add more dogs in shelters. There is a lot that goes into breeding, so if you do decide to take that route, I strongly encourage you to do your research. If you are interested, check out the Functional Dog Collaborative website and podcasts. They are wonderful.