Finders-Seekers, by Gayle Greeno seems to match a number of elements that you remember.
Their technological resources destroyed, a colonizing expedition from Earth has been stranded on the world of Methuen for over two hundred years. Their continued survival is largely due to the organization of healers known as the Eumedicos and to the Seekers Veritas, a unique group composed of pairs of Bondmates, one human and one ghatti—a telepathic catlike being native to Methuen who bonds with a specific human for life. These Bondmates travel from town to town, settling disputes by truth-reading the minds and emotions of plaintiffs and defendants. While most people respect the Seekers, there are those who fear the ghatti powers. And now someone has begun attacking Seeker pairs.
What no one knows is that this destroyer has targeted one specific pair of Bondmates as special victims—the woman Doyce and the ghatta Khar'pern. For the key to defeating this deadly foe is locked away in Doyce's mind behind barriers even her ghattas has never been able to break down.
This seems to hit on a number of your main points. The cover has a girl and an orange cat. (There's also a gray tabby cat, which is a strike against it.) The girl and the cat are telepathically linked. The setting is a colony planet that's lost its technology, so that's a possible match to your post-apocalyptic setting. The woman and her cat are being targeted and hunted.
Unlike the book you remember, the main character of this one seems to be a woman, not a young girl. And the summary blurb doesn't mention experiments or doctors.