not to mention, historically the kingdoms of Hammerfell and Skyrim have rarely ever been "buddy buddy".. i wouldn't be surprised if such an alliance just created further unrest among the civilian population
if they did though, hypothetically speaking, I still don't think they would have the forces to take on the dominion on their own turf.. one of the main reasons the Redguards were able to repel the Dominion was because of the enviorment, the dominion just wasn't prepared for the environment of Hammerfell..
similar to the reasons Hitlers attempt at invading the Soviet Union fell apart, his people just weren't prepared to fight in such a climate (funny fact although somewhat off topic, at the time of WWII many of the allies military commanders believed the Canadian military as the only ones who could possibly successfully invade the Soviet Union, because of the similarities in climate)