On the substance of your question, it may well be that different philosophers will respond differently, though I'd guess that naturalist/determinist/compatibilist more often goes with a view of punishment as having broadly utilitarian goals rather than retributivist ones. But I was struck by your last sentence: you find it disturbing that compatibilists would be willing to punish people whom incompatibilists see as innocent. Isn't this really just a way of siding with the incompatibilists? Compatibilists argue at length that we can be morally responsible even if determinism is true. Indeed, some compatibilists have argued (Hobart is a famous example from many decades ago) that we can't be responsible unless determinism is true. If compatibilists can make their case, then their point of view is only superficially disturbing. The apparently disturbing character, they would argue, is an illusion borne of misunderstanding what's required for moral responsibility. The compatibilist, in other words, thinks there's a strong case that these "innocents" aren't really innocent. Whether they have the better case is, of course, a matter of debate. But they do have arguments to offer.
Read another response by Allen Stairs
Do philosophers who believe in a naturalistic and deterministic world and assert a compatabilist theory of free will believe that people who do very wrong things should be punished as an expression of retribution or to make the person realize how bad they are? (Rather than the use of punishment as discouragement) I find it fascinating and deeply disturbing that philosophers would want to punish people who are perfectly innocent according to a incompatibilist ethical system.