If it was already in the state of death gasping, then you essentially were hitting an already "out of it" animal. The cat's body had already gone into shutdown, and it had no awareness that it was suffering.
It seems more a case of relieving your daughter's belief that the cat was suffering, and you are to be commended for taking action to ensure her that the cat would no longer be suffering.
As far as it being judged a violent form of euthanasia by you or your daughter, that is ultimately between you and your daughter, as to how she ends up being effected by the "shovel to the head" method itself. I have a hunch, however, that society need not worry that your daughter will become a serial killer, because I have the impression that you deal with realities with your children from a very level-headed standpoint.
otoh, when she's a teenager and you end up grounding her once too often for staying out too late, I'd play it safe and wear a helmet every night when sleeping. Okay...j/k.
