He, however, sees a narrower window because of Special Relativity, and assumes that the mouse will be saved by the meter stick hitting the window.
The mouse sees a short meter stick, and assumes that he is doomed.
Who is right?
From the cat's point of view, the sensible way to push the meter stick is to apply an impulse along its length, applying it simultaneously in the cat's (and rocket's) frame of reference. This makes everything that the cat said correct.
To appreciate the mouse's view, you need to construct the correct spacetime diagram of the scene in the mouse's frame. Here is an image of such a spacetime diagram.
The gold strip is the world strip of the window opening. The mouse is sitting in the middle of this. The blue strip is the meter stick in the rocket before it is pushed. The red strip is it on its way to attempt to hit the mouse. Note that it overlaps the window on both sides.
To see what is going on, look at the green slice, which is a slice at constant mouse time. In this slice, the meter stick appears tilted. A ha! That simultaneous push in the cat's frame was not simultaneous in the mouse frame.
Here is an image of the spacetime diagram using Geomview so that you can rotate, enlarge, and whatever with the image. If you right click on the different world strips, you will bring up explanations of those parts in your browser.