How to fix a garbage disposal with rubber caught in it?
Reviewing the design of a garbage disposal here it seems very difficult for anything like a chunk of rubber to actually get jammed in the grind ring.

If you were to put something in there like a chunk of rubber (or bone), it seems likely it will bounce around and not get ground up at all. Small bits or rubber might get broken off in the ring but by the design they should be washed down the drain. I have experience with the piece of bone thing and it does, in fact, just bounce around irritatingly refusing to be ground up.

It would seem the worst things for disposals are slimy sticky things that are not water soluble. These materials could get into the grinder ring and clog it up similar to a file being clogged with paint.

So, if your piece of rubber was batted into the disposal by your spiteful cat, it is likely still in there bouncing around. Your choices are: have a child or slender-wristed lady fish it out; or remove the disposal, dump it out and possibly disassemble it to clean it out.

BTW you may want to keep the cat a little happier since you do have to sleep sometime.