To the specific cat example you gave, I'd say have the cat euthanized (however hard it is, is there anything to gain when you'd experience grief and suffering anyway seeing your lifelong companion in agony?) I don't feel that this example is typical of the situations in the argument topic, however, because I don't believe any feelings would be spared.
However, in general, I feel that it is a very legitimate decision to save your own feelings (where they can be saved) and not to undergo emotional pain, depending on what the stakes are for the other beings in question. To make it immoral to save your own feelings opens the door to becoming a martyr to the whims of others and their demands. Ultimately we need to love and honor ourselves first, and then love and honor others.
There are always the exceptions, like your children, but if somebody is going to emotionally hurt in a situation, why should it be you? I get it, this sounds selfish, but that's what humans are right? There's nothing wrong with not wanting to be hurt.. We as humans naturally shy away from such things, which is completely normal. There are of course instances, like the one about the cat, where if the thing is in physical pain, ending the cat's life is better for the both of you in the end, because you will undoubtedly suffer from watching your cat suffer. Doing things so you hurt and others don't can cause you to look like a pushover who will take everything, even if they don't deserve it, and this can cause you even more pain in the future.
Firstly, I'm sorry if this is something which is personally affecting you (to the person who posted), it's a tough thing to go through. However, I believe the right thing to do is what is best for the animal and put an end to their misery. Quality of life matters, being riddled with disease and crippled with pain removes all traces of that. This is precisely the reason I am for legal euthanasia. When you love someone, or a pet, you want to do the right thing by them, that's not always going to coincide with what is best or easiest for you. Love and compassion require courage.
Here is a dilemma, just to clarify my question a bit more-
Your cat has feline leukemia, and the vet says it is fatal. Since the disease will cause a lot of pain to your cat, the vet suggests to euthanize him in order to relieve him of suffering. Even your parents agree. However, you have know your cat ever since a toddler, and have grown up with him, playing, cuddling, whispering your secrets, etc. He has been a lifelong friend whom you can seek comfort. You really don't want to see your cat just leave you, and want him to stay with you as much as possible, until his last breath. On the other hand, you can't bear to see him unhappy and suffering . Is it moral to let him stay with you, so you can spare yourself some grief?