Going back to the original idea of relativity — simply looking at the relative motion between objects, and the idea that you are allowed to look at the physics in the reference frame that makes the analysis the easiest.
The other thing [besides analyzing momentum] you can do is to invoke relativity a la Galileo. The problem where both bat and ball are moving is still kind of a pain, mathematically, but if one of the two is stationary, then your life gets a whole lot easier. And we know what happens when a light moving object like a baseball hits a heavier object like a bat that isn’t moving: the ball bounces back at a fair fraction of the speed it came in at, and the bat only moves a little bit.