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I’ve been attempting to obtain the rank for the past week, and obtained it after quite a bit of research and attempts. So far, I’ve found conflicting opinions from Bulbapedia, Serebii, Reddit, and GameFAQs. I can confidently confirm that the requirement is not solely to defeat every individual trainer, as I have already tracked myself doing so. I’m thinking that it is a combination of a hidden number of points and to have defeated every trainer, but I am still unsure.

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It's definitely not a matter of defeating every trainer. The Chateau logs your total points (though you can't see them, from what I remember), and that's what causes you to rank up. Unfortunately, it doesn't seem like Bulbapedia knows what the Grand Duke/Duchess point threshold is.
I remember absolutely grinding the Battle Chateau when I was younger. From what I remember it took an absurdly long time to get to Grand Duke
From what I remember, defeating every trainer is definitely one way to do it, but the most efficient way includes focusing on the higher ranked trainers (as they give you more points) and using specific Writs (don't remember exactly which, but it's likely the most expensive ones). Though most character models are used for all the ranks, there's a higher concentration of certain models for certain ranks. The Bulbapedia article you linked has them all listed. But definitely, the Writs are the game breaker here.
How do you know this for certain? Also, from my question's description, I beat every trainer, but that did not give me Grand Duke. The question asked for the "exact method," but you gave a vague guidance.
I just got Grand Duke on Y, and recorded my points as I battled. I recorded 616 points, but I only started recording after about 100 points after I got Duke, which leads me to believe that 1000 points is the requirement, since 300 is the requirement for Duke. However, I also recorded myself beating every possible trainer before I reached 616, so I believe that it may be a dual requirement.

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