I hope you shot the grey squirrel - or North American tree rat, as I like to call it. Now the rats, they're preety indigenous (I think they came over with the Romans).

How about hanging the fat ball from a high (as high as you can (safely!) reach on your stepladder) and thin (too thin for the cat to climb along) branch of a tree.   Or you might try and contrive something with bits of string strung from an upstairs window out to a tree, and a vertical one hanging off that...?   No?    Maybe not.

Just an idea.

S