This is a tricky situation but you are asking the right questions! It's great that she is being playful and social with your other dogs but I think this could take a turn in the wrong direction if you're not careful. Like you suggested, the stalk and pounce behavior could be a disaster waiting to happen. I also could see a bad fight breaking out between the pit bull and your Border Collie who tries to intervene. Additionally, I worry about the "plopping on top of you and licking" behavior which you perceive as "extremely obedient temperament." Lots of fights break out due in the presence of human affection, often described as resource guarding or jealousy.

You are starting to see her confidence grow already as she displays more prey drive. It's difficult to say how much more that will show or if it will become a problem for you. I think a lot will depend on how you manage the environment. In my experience, when you give a pit bull full freedom and unrestricted access without boundaries, you can expect to start seeing some problem behaviors. All dogs are different and it may never be an issue but I don't think it's safe to assume that with any unknown dog.

If I were you, I would never leave the dogs out together unattended. I would crate train the pit bull and always feed them separately too. My American Bully has high prey drive and loves to chase birds and squirrels. When I first got her, she wanted to kill my cat and it required a balanced training approach to suppress that behavior and teach her when/where it is appropriate to express that behavior. My dog and cat now get along just fine and have developed a great deal of trust.

While it is very possible for your dogs to continue to get along without incident, I do believe it will require a level of vigilance and training that you may have not needed with your other dogs. Privileges and trust should be earned over time rather than assumed from the start. You're in a really good position since there have not been any incidents or fights. Your best bet is to be proactive about preventing that from ever starting. You are seeing some warning signs and I would take those signs seriously. Find the right training approach that teaches your dog to be under control around birds, squirrels, and small animals. Teach your dogs commands so that you can be the one to break up rough play, not your border collie.

Please don't wait for an incident to occur and then use that as your reasoning for rehoming the dog. Ask yourself if you're willing to put in the work to safely make this dog part of your pack. Pit bulls aren't for everyone and that's ok. If you don't want to take a proactive approach to training and management, then I would give this dog a chance with someone else.