It's a tricky issue, as is taking the life of any entity that has the ability to feel for that matter. There's no way for us to know what type of pain is incurred upon death by the animal, we can't talk to them in the first place, and even if we could, the ones we put down wouldn't be able to say anything. Even if the process is physiologically painful, the psychological process of death can be frightening. In humans death is generally accompanied by the release of dimethyltryptamine (DMT) by the brain, it's an extremely potent hallucinogen. There are lots of case reports that can be read from people who take DMT recreationally, and they run the gamut from beautiful experiences that one would be happy to have as their last, to horrendous traumas that one may never recover from completely. I don't know whether other animals also release DMT when they are in the processes of death, but if they do it's hard for us to know how the chemicals used to put an animal down would interact with DMT, or any other chemicals released by the brain upon death. The point is that even if the physical pain of putting an animal down is negligable, we can't know what it endures psychologically.
That said, I do think that it is generally safe to assume that the euthenasia of an animal which is suffering may be a greater mercy than allowing its life to run its natural course, though this too is a tricky issue to judge because while it is easier for the animal, it is also easier for the owner. That is, we project our own feelings onto the animal, when this may or may not be acurate. Still, I imagine that the pain an animal endures from an injury that will ultimately kill it is greater in some ways than what a human with similar injuries might experience. Unlike humans, whose brains permit them to wrap their heads around their own inevitable deaths, and in the case of some to even accept their own mortality, it could be pretty safely assumed that concepts like life and death simply aren't something an animal can apply to its own consciousness. For an animal who is suffering, they do not understand that the pain will be terminated, even if its termination comes in the form of death. This means that unlike humans, whose brains permit them to see a future in which pain does not exist, for an animal pain is an all encompassing feeling, they lack the ability to swim through the pain, and instead drown in it. To this extent, it is probably much more merciful to end the life of the animal instead of allowing it to continue suffering.
That said, these are all arguments phrased with utilitarian logic. That's all fine and good, but ultimately the decision may not be utilitarian at all, but deontological in nature. I do not claim to know whether or not there is a God or an afterlife of some sort, but postulating the existence of a God makes all utilitarian logic null. If a God does exist, then the decision of whether or not euthanasia is morally appropriate is not something that we can judge for ourselves, but instead is something we must be told by God. I do not know the OP's religious view, but I suggest that if the OP has religious views that they consult their own metaphysical beliefs in order to best judge whether or not the euthanization of the cat is for the best.
