How to involve children in clicker training the family pet? Some children of eight or nine grasp the principles the moment they are exposed to them, and can go right to work; having no preconceptions they catch on a lot quicker than adults sometimes. Three years old may be a bit too young for the insight into cause and effect that clicker training takes. By all means she can click her stuffed animals or pretend animals; but I'd suggest that with the cat and the dog she be allowed to click and give treats, three or four treats per animal, once a day, say. That will be clicker training, as far as she is concerned; she'll understand more about it as she gets older. Meanwhile the dog will incidentally learn to sit and hope for treats from her, and the cat might learn the same.
When a cat is afraid of a young child, I suggest that the child be dissuaded and distracted from approaching the cat in any other manner than with the click and treats. That the cat is now afraid of her suggests that some gentle clicker training of the child is in order, to alter her behavior toward the cat. Kids need to learn to pat gently, to approach animals politely, to avoid sudden movements, not to grab, poke, or strike at animals, and so on. Toddlers need to be supervised around animals at all times, but they can learn good animal handling. My three-year-old grandson, Nathanial, is often in my house and learned all these animal skills much earlier than three. You can see him on my most recent video, Puppy Love, at 18 months, planting a kiss on my 12-week-old puppy's head, very carefully, mostly taught by his older brothers.