Should I 'exterminate' this cat? Is it even legal?

I think you could catch it with a larger live trap. Put some meat in there as bait. It might take more than one day. You might want to put a blanket over the trap and put in under a table or something. Try to hide it. Animals are smarter than what you think when it comes to traps! I grew up in the country with mean tom cats all around that hared our cats and killed kittens. We used traps many times to catch them and take them elsewhere.