I'm legit surprised there isn't a mod like this out there. ES modding has a proud tradition of making mods that let you join the bad guys or at least allow more of an 'evil path' for your character. Morrowind has Great House Dagoth, allowing you to play an alternative main quest by joining Dagoth Ur, and Oblivion has Servant of the Dawn, allowing you to rebuild the Mythic Dawn after the main quest.
I'll start off by just saying: I don't know if I could actually make this thing myself. At most I might be able to write dialogue/quests, voice NPCs, and maybe, MAYBE code, but I'd need to learn. I mostly just wanted to write this down for the sake of posterity at least.
So, I've several different ideas on how the mod could possibly come about. The ideas are not related, and if hypothetically this mod were to be made, I imagine only one idea would actually get made.
1. Alternative Main Quest: Immediately after Helgen (or after loading in from an alt start mod), you are given a mysterious note by a group who knows what you are, and to travel to the ancient city of Bromjunaar to find out more. From there you go to Labyrinthian, find your contact, who then takes you into a new area of the ruins where a small surviving splinter faction of the Dragon Cult survives, and they tell you that you're Dragonborn (even though you've not absorbed any dragon souls yet). From there begins an alternative main quest, which hits a lot of the same story beats, but trims down on the fat a lot, as the cultists would have much of the knowledge you would need to fulfill your destiny AS Dragonborn, without having to run all over creation as much just to get to fighting Alduin. The twist is perhaps it is a bit more of an 'evil' version of the main quest, for example, instead of being given 'whispers' like you do for the Greybeards, the Dragon cultists can directly teach you words (but not unlock them) for a filled Black Soul Gem.
2. Just another faction: Basically it would be just like any other faction. You could join it, do stuff for them, rise in ranks, etc. I'm not entirely sure on a storyline for this idea right now, but the big end reward would be a new Dragon Priest mask, different from all the others, or perhaps even a new Shout, or something along those lines. Can also see some quests being related to gathering items from the Dragon Cutl's past, such as the other masks, or new items added by the mod.
3. Champion of Hermaeus Mora: After Miraak is defeated, you return to Solstheim and begin setting up the continuation of the work he left unfinished, but now it shall be about you as the truly LAST Dragonborn setting up your rightful place as the ruler of Solstheim. Bend the will of the All Maker stones to obey you now, and spread your influence by collecting the Black Books, to build a temple to your supremacy, with the island's inhabitants enslaved to your will for all time.
Anyways, the ideas are vague and open I know, but I more wanted to just put them down at least while I thought about them for posterity. Honestly if it were possible, I'd love to see this get made in some form or fashion, even if none of my ideas are used. I also think it'd be fun to see some integration of other mods like LOTD, Konahrik's Accoutrements, and so on. Could also see part of the mod turning the overworld part of Labyrinthian into a restored and functioning state, making it into a viable player-made/run city with services and the like. That's all I had, and so yeah, just wanted to make mention of it while it was on my mind.