I’m really sorry to hear that.
Unreasonable people require different approaches. You could ask your biological dad if he’ll support you in taking the dog, and ask him what he thinks the most effective approach would be. If he got you away from her, maybe he can get the dog too, except he’s not the dog’s father, of course.
You may be right that it would involve a “custody” fight, which I think most places would be a small claims lawsuit, treating the dog as property. If she shows up, the judge would ask you and her questions to try to figure out where the dog should go, but if she’s the registered owner and you can’t show that the dog is suffering at her house, it might be hard to win that.
The other angle would be your understanding of her nature and choices. If you can figure out what she actually wants and cares about, that you have the power to give or deny her, then you might be able to bargain with her. Does she want anything from you that you’ll be able to deny, such as being willing to show up for a holiday, or anything you are able to take with you that she wants?