Snap it’s neck in one of the many suggested methods, this probably only paralises it but unless you check it’s heart beat…….short lived though, i usually hold it by the rear legs and chop down on the back of its neck, if they’re too wriggly then i crack they’re neck by force.
All depends on how i feel at the time.
A snarling injured Badger would be hard work to dispatch, but i’d prolly use a swift blow to the back of the head, not easy when it’s squirming around, never leave an animal to suffer unless it’s too dangerous to deal with it(roads).
BTW, a badger will turn in it’s den and block then entrance with it’s rear, a dog would need claw at it to get to the cubs.
Proper ‘ard b’stids!