Hello, my name is ***** ***** I will be helping you today.
Unfortunately, unless there is indisputable proof that the horses are yours, the local officials in Georgia will not get involved. She is probably saying that the horse is rightfully hers, while you are saying the opposite. The police won't arrest someone if they do not have clear evidence pointing to a theft. That is why they are saying this is a civil matter. (although there is no reason they should tell you to call your Wisconsin police; that does not make sense).
So that you will need to do is actually bring a lawsuit against the woman to reclaim the horse. You will probably have to bring this lawsuit in Georgia. While this will be a pain since it is so far away, because the woman really has no contacts in Wisconsin, you are unable to bring the lawsuit there. And even if you could, the court would probably say that the case makes more sense to be brought in Georgia since that is where the horse is and require you to transfer the case.
One thing you will have to decide is whether it is worth the cost of bringing the lawsuit and seeing it through in Georgia. As a non-profit, I'm sure expenses need to be somewhat limited. Even if you end up winning the case, that just means that the horse needs to be returned to you. It does not mean that you get to recover fees and costs for bringing the case.
My wife and I actually own in OTTB and it's great what you are doing. I'm sorry that this has happened to you and that the legal system might just be too costly to handle this situation.