Really, I'm not trying to be a jerk here, but...are they paying you or are you paying them?
If they're charging you $10 so that they can snuff these un-adoptable cats, I must wonder why you would pay? A careful shot is virtually free by comparison, and is more humane. Catching a wild animal in a live trap, then toting it in to the H.S. must be a horror show to the critter. If you are truly considerate of it, then shooting it is the obvious answer.
If they're paying you, then I need to know if it's a dead-or-alive deal.
Canuckgunny, you're blunt, but you're right. However, I disagree with the coyote bait comment...I find they work great. They also disappear quickly from my vulture feeder: a 4x4 foot piece of plywood mounted horizontally on top of a pole out in a field...but if the H.S. is paying a bounty on'em, the vulture feeding ends now!
John