The principle of inflicting suffering on moral criminals even outside of prevention and deterrent - is one of the most important pillars of Protagonist morality. This is unfortunately one I cannot provide any logical argument for, because the most important things about morality are not the things that can be proven. They are the axiological things, the things that must be learned through conscience and not reason. Retributory Justice is one of those. So I will not give an argument for it. I can, however, refute a few of the common objections to it.

1: "Punishing people for their wrongdoings doesn't make them good people." (Perfect Solution Fallacy)
That's completely true. You know what else is true? Feeding starving orphans doesn't change the aspects of human nature that create poverty, so let's not do any of that either. You see the problem? For any good action you can name another good objective that it doesn't accomplish. No action furthers every good objective. So just because there's one good thing punishing unrepentant criminals doesn't do doesn't excuse us from the duty to do it.

2: "Punishing people is sometimes okay as a deterrent for future wrongdoings, but to do it just for the sake of some abstract idea of 'Justice' is pointlessly hurting people." (Circular)
You have a disputed assumption here that it's pointlessly hurting people. My claim is exactly that it's not pointless, so this argument is circular. Our real difference is one of axiology, which can't be logically refuted unless it can be shown to be internally inconsistent.