Manslaughter would be more achievable. Even then, you'd likely have to prove recklessness.
Surely if you've allowed your dog to get into a situation* where it's able to get its jaws around someone, that's by very definition "reckless"?
* Obviously ignoring scenarios outside your control e.g. where someone has climbed a fence and is trespassing on your private land where the dog should be expected to be able to safely roam free.
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