Well either the other clans would simply be absorbed, with the ilClan "winning" the grand game of clan culture. More likely they'll at best be like new march lords, at worst integrated military commands of the new Star league like the SLF corps or Avalon Hussars.
reform the SLDF, remove the corrupt house lords and their evil taint, freeing the masses from what the clans would call slavery and evil oppression of the house lords, to become bondsmen of the clans (they don't view bondsmen as slaves for some reason). The clans would become the military and political advisers of the new SL.
Well, the clans would mostly let them keep doing what they're already doing. Merely rebranding them. Farmers would be "laborers", law enforcement forces that capitulated would be folded into the Watch. The worthy candidates of the IS armies, would be allowed a chance into the Touman as second line troops, garrisoning the worlds they once protected.
What could change the paradigm is if the Clans conquered Terra and got their hands on actual Star League records that would destroy the Clan mythos of the Star League and its purpose, which was the very opposite of Clan doctrine of conquering the Houses, but working with them as equal members. Heads would explode.
That's a fallacy. The clans didn't need to find records about how the star league worked. Their records were far superior and complete than those in the IS, which is why post-invasion IS scholars would turn to clan loremasters to clear out the mess. The clan had a clearer view of the star league, for all her faults; it wasn't tinted with myths about a magical fairytale about how everyone worked together and universal peace. They're not nostalgic about the SL, they've judged it flawed and wants to build a new SL based on the clan system.
The clans definitely had the muscle to dominate interstellar travel. They theoretically could smash the battlemech forces of the IS. Garrisoning thousands of planet would've been a problem. But look at the clan OZs, especially in the DA. they seem to be handling themselves well. Now imagine instead of 4-6 clans, there are three times as much. They would've no problem dominating the majority of the IS, especially under the authority of the ilclan stopping them from fighting each other and stumbling on their toes.