Can basic desert and retributivism be justified under Compatibilism?

From my research into the topic of free will I can support Compatibilism and the idea that the kind of free will worth wanting is where you’re free to act on your desires. Despite this I’m having difficulty understanding the moral responsibility and punishment part espoused by the likes of Daniel Dennett specifically the idea that people genuinely deserve to be punished for doing wrong for reasons other than deterrence, rehabilitation and incapacitation.

I read and watched Dennett’s discussion with Gregg Caruso about free will and Dennett often speaks about the “Moral Agents Club” and how if you want to live in a society and enjoy its benefits you have to be held morally responsible in a similar sense that people play by rules in a game and by doing so subject themselves to punishment when they make a mistake or lose. He uses the analogy of getting a red card in soccer. It has to work that way otherwise the “game” of society collapses and ceases to function properly. At the same time Dennett rejects backwards looking basic desert and retributivism. Overall Dennett’s view seems more a matter of practicality than what laymen truly mean when they say someone who has done evil things deserves to be punished or express satisfaction when something bad happens to a wrongdoer. To quote someone else desert without retribution is just another name for attribution but we don’t need the concept of free will for attribution. When someone says a serial killer should be executed or it’s good that a villain gets killed by the hero in an action film they almost certainly don’t mean it in Dennett’s red card/Moral Agents Club sense. They mean that the person genuinely deserves to be punished for its own sake regardless of society’s laws or some imaginary social contract. To them it doesn’t matter if an evil person was punished in a normal society by a court or in a barren desert by a vigilante. Moral desert is moral desert in whatever context. Immanuel Kant used the example of a murderer being executed on a desert island once the society has dissipated and the citizens go their separate ways.

With all of this in mind are there any sound non consequentialist arguments for basic desert and punishment under Compatibilism or are the ideas simply too irreconcilable to be held simultaneously?

Are there any good sources on the matter that can help me understand the issue?