The thing is that Paul could have married Chani. By the end of Dune, Paul claims to be fully culturally Fremen, and in Fremen culture, men can take more than one wife. No Fremen would have raised an eyebrow at Paul marrying both Irulan and Chani...
But the Great Houses would.
An argument could be made for Paul marrying Irulan and not Chani because his marriage needs to be political and having another, Fremen, low-born wife might jeapordise that, but it’s not like he ever has much time for Irulan’s thoughts and feelings, and his political position at the end of the book is pretty secure. Sure, a second wife would be an insult to Irulan, but not much more than the fact the he ostensibly kept a lover while it was known that he never even touched Irulan.
It’s also interesting that marrying Chani is something that doesn’t even occur to Paul in his internal monologue. It’s never an option in his mind. Paul thinks monogamously. For all that he claims to be culturally Fremen, when it comes to marriage, he still thinks and acts more like a nobleman of the Landsraad.