The options that cannot be used are shooting (because of legislation) and lethal injection (because of expense and need for trained personnel onsight).
I have seen poison baits administered by municipality staff to stray dogs to control their number. I am not sure about the poison, but they used meat as a bait and it is quite effective. I have seen dogs passing out in a couple of minutes and dying ultimately.
Based on ecological theories, the most humane and cost efficient method is to let the nature control their populations.
killing non target wildlife species or pets is a big concern when trying to remove feral dogs and cats. Do you think you might be able to suggest a two step program : first live-trap, second selective euthanasia?
Regarding the second step, would the use of large killing traps be a possibility? (I would require opening the live-trap cage at one end of a tunnel, and setting the killing trap at the other end, or a similar approach). Some traps seem to be designed for relatively large species (cf http://www.fur.ca/certifiedtraps.php). The question of how humane these traps are is debatable, but they seem to result in a very quick death when used appropriately.
I'm sure you're already aware of this, but you might want to consider the fact that removing feral animals may result in demographic instability, and may not be the best solution depending on your problem.