Killing Fluffy the Mining Poodle and eating her would likewise constitute self-defense of the hypothetical miners' health. Not eating would lead to muscular and systemic weakness, preventing the miners from escaping the cave-in.
Killing is not a sin per se. Killing can be a holy act -- for example, killing a robber, a rapist, or an enemy soldier in a war. Killing an animal for food is perfectly moral, but such killing should be done as quickly and painlessly as possible -- not because the animal has any rights we need respect, but because a callousness towards suffering tends to kill one's empathy towards human beings. The act of killing a dog or cat (or any other animal that trusts people enough to form an emotional bond with its owner) isn't itself a sin, but it's a sign of a person who has no respect for loyalty -- a sure sign of a sick soul.
According to your theory, it would also be an act of "self-defense" for one of the miners to kill the other in order to survive. Not eating is fasting, and that won't kill you for a long time, but dehydration will kill you very fast. I don't think it's an act of self-defense to kill a trusting animal just because you're smarter, you're "entitled" to survive, or just because you can. Otherwise, to paraphrase C.S.Lewis, as a matter of self-defense, we should kill the Democrats as a matter of self-defense.
If we cut up beasts simply because they cannot prevent us and because we are backing our own side in the struggle for existence, it is only logical to cut up imbeciles, criminals, enemies, or capitalists for the same reasons. C. S. Lewis