Sedated with a dart? By injection if a small trap, just poke the needle in.
Maybe the cat will come over and sit on the needle for you. Sorry, just being facetious.
Have you ever caught a cat in a trap? If so, you'd know what you were dealing with here. This is not the cuddly little ball of fur that sits by contently in your window or on your hearth. This is a wild animal, and wild hardly begins to describe it. Just approach that trap, and you will immediately see a snarling, spitting, clawing, ball of fur, that would just love nothing more, than to take your eyes right off of your face in a very, non-humane manner. This animal will reach right through the cage and inflict great bodily harm to you, and it will do it so quick, you won't know what happened till you see the blood. But let's say you manage to get it to take a sleeping pill, or whatever, and then you have it neutered. Well now what? That animal is still going to be wild as a Tasmanian Devil, and it is still going to remove your eyes, given the opportunity, and it is still going to prey on all the same innocent wildlife, as it has always done.
As amc rulz stated, all "Great ideas, just not practical. We're talking thousands of feral cats, and there's lots of people out there that deal with the problem in their own way, in much less humane ways than an arrow through the head, sad to say."
Again, I'm not taking her side (the vet), but unfortunately, the feral cat is something to be dealt with, one way or another.........it's a real problem, and until you have cornered one, face to face, up close and personal, you'll never understand what you are dealing with.