You shouldn't be using Lorentz transformation. The time difference for part c) is not just a Lorentz transform of the time difference for part a). This is because the events representing the 'initial' spacetime coordinates of the balls are different for parts a) and c). In other words, its not just a change of frame of reference; we are also talking about different events.
There is another equation you should use. In the rest frame, you know one ball is stationary, so what do you need to find out about the other ball to get the answer?
