I'm in central London quite a bit with regard to shooting foxes, just to get the record straight here are some hard facts..
(1) make sure you have insurance (for firearms)
(2) .22 Rimfire is my choice and its almost ALWAYS an elevated shot (forget the .17 unless you want it to be your last job for the client)
(3) Almost all of this work is covert, discretion is the order of the day
(4) YOU decide whether or not to inform the Police your not obligated to, i've had clients that have insisted we dont because of the publicity and believe me if you get through to the wrong sort its