However, a very similar story has been proven true (or so Wikipedia claims): https://en.wikipedia.org/wiki/Great_Hanoi_Rat_Massacre
In what scenario is it easier to cut the tail of a living (fighting) rat, than to kill it first and the cut the tail off a dead rat?
But if your goal is to maximize the number of rat tails you end up with, cutting it off a live rat is probably worth the additional effort.