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.