I think it’s important to note that Dean killed Lucifer as Dean and not AU!Michael. An angelically powered Dean, sure, but Dean none the less.
Not only is killing Lucifer a Jesus thing (the most Jesus thing Dean could ever do, really) killing Lucifer as Dean doubles down on it. And why? Because Dean is the epitome of Free Will and Humanity. He’s The Son of Man, if you will.
AU!Michael, on the other hand, is the archangel who destroyed AU!Earth and made himself God in the total absence of God and aims to do the same to OG!Earth. He is, for all intents and purposes, the AntiChrist.
Dean, being the bastion of Free Will that he is, will fight Michael for as long as Michael possess him.
So while I’m less than thrilled about Michael!Dean, we could very well get Dean and Michael duking it out inside of Dean in a sort of internal battle of Armageddon.