Hmm. The most prominent opposing brothers in US political history were Bob and Alf Taylor of Tennessee. They ran against each other for governor in 1886; Bob (the Democrat) won, 55%-45%. They appeared together at debates around the state, and often held a joint fiddle concert afterward. If the local hotels were full, they sometimes shared a room and even a bed.
Both were active in Tennessee politics for many years. In 1910, Alf ran in the Republican gubernatorial primary, losing to Ben Hooper, who then defeated Bob in the general election. In 1920, Alf was finally elected governor as well.
Other than them, I draw a blank. There have been several political dynasties which had potential president brothers, but none with a party split. One brother would have to bolt to the other party.
Maybe Winthrop Rockefeller could flip to Democrat after moving to Arkansas, and somehow end up running against Nelson.
EDIT: curse you, David T.! Beat me by 25 minutes!