Killing rabbits with a looting III sword doesn't give the same increased drops for meat as other animals.
For one test in 15w51b I killed 36 rabbits with a looting III sword and received 71 rabbit hides, 21 raw rabbit, and 6 rabbits feet.
Using an clean sword and killing 36 rabbits resulted in 15 rabbit hides, 14 raw rabbit, and 9 rabbit feet.
So while looting III significantly increased the amount of hides received the same can not be said of the meat.
This is true (can look directly at the loot_tables), and is different from all the other food animals, so it may indeed be a bug/oversight.