Yikes! What a situation! I love cats, but NOT their pee on anything in the house. I wonder what that cat was reacting to when it smelled your quilt.
I do like the idea of inserting the iron on batting in the area that has thinned out. I've actually had a situation where I had to do that, (not from a cat!), and it turned out just fine.
I would make sure the door to your sewing was shut from now on...........can't let that happen again!