Lol I'm kinda with you. While I obviously don't love the idea of killing toads, but in a war against invasives you gotta get your hands dirty! And I agree; I would say an instantaneous death is a humane one (when we feed our reptiles mice, we kill them instantly to be humane). I know they used to freeze the toads en mass, but it has been shown that freezing to death is an unpleasant death as cells burst when freezing (they don't just 'drift off to sleep' as people once believed!).
I'm not sure about the foxes. I'd imagine a similar effect as they have had to overcome the invasion of the toads as well, though I cannot conclusively say for sure!