I still think there is a problem with the concept of "humane slaughter" or "compassionate slaughter" of animals, let alone breeding and confining them with the aim to slaughter them for meat eventually. Even a "compassionate" farmer who practices free range and organic farming still slaughters his livestock after allowing them a very brief life. How can this be called compassionate? I think euthanasia is the only form of compassionate killing. And a young animal in prime health is not eligible for euthanasia. Is a farmer still compassionate if he treats his livestock kindly before killing them?
How would one define or characterise a compassionate omnivore? I'm not rejecting this idea out of hand, but I would like to know what it means. Early human and pre-humans were not predators but scavengers who ate the remains of carcasses once the true predators had finished eating or been chased away. It was only once tool use became more sophisticated that humans (and Neanderthals) became hunters. I think our very tool use (prosthesis or technology) compromises our compassion. Perhaps a compassionate omnivore is one who eats road kill? But once animals are confined with the aim to be killed, I don't see much room for compassion, although there are various degrees of cruelty (and perhaps kindness) in the ways in which they are confined or treated while confined.
So, what would a compassionate omnivore be? I can understand that meat eating may occasionally be necessary for medical or survival reasons, but being necessary doesn't make it ethical, still less compassionate. Our sciences are so far advanced today that we can design much more compassionate (vegetarian and vegan) diets. Yet we use science and technology to prop up factory farming (increasingly intensified confinement, growth hormones and steroids, antibiotics to try to combat the higher levels of resultant disease, genetic engineering, etc.). Thus, science and technology are used to intensify cruelty rather than achieve compassion. My hope is that science and technology will be used more compassionately in future, which is contrary to how it is used currently. I suspect our survival as a species will depend on that.