If the Dhovakiin sides with the Stormcloaks, then the Dominion has to fight against a united Skyrim on the victory rush of defeating the Empire (Giving them something similar to what Tiber Septim's Empire experienced in it's quest to conquer Hammerfell once Cyrus kicked the Empire's ass at the Second Battle of Stros M'Kai), led by Ulfric Stormcloak and the incarnation of Talos - who may go on and conquer all of Tamriel again.
I don't know, I have a hard time believing that Skyrim could stand alone against the Thalmor and win, Dhovakiin or no. A united Empire (Skyrim included) barely managed to hold them back, and Skyrim would be weak following a bloody civil war to oust the Empire. A successful rebellion would likely end up shattering the already fragile Empire, allowing the Thalmor to steamroll across Tamriel. Dhovakiin may be strong lore-wise, but he's not a one man army. Even if he and an Empire-free Skyrim resisted the initial Thalmor invasion, the rest of Tamriel would be swallowed up, and the Thalmor would only grow stronger. At that point it's only a matter of time until Skyrim falls, too. And honestly, playing through this game I haven't ever gotten the impression that Skyrim is any sort of major military power, most of it is pretty back-woodsy, with only a couple major population centers. Where would Skyrim marshal the military strength to stop a major invasion by a powerful army, after ousting or killing most of it's trained fighting force (Imperial soldiers)?
It just seems to me that joining the rebellion is hardly a good bet for anyone interested in restoring Talos worship. I'd say Ulfric is shortsighted in this regard, but as others have pointed out, the Talos issue is only a pretense. The man is just a racist power-monger with a personal agenda.
Edit: Beaten by the guy above me.