I don't think that a dispute would have ensued, if both twins had remained alive. The Rohirrim all led quite dangerous lives, and they were prepared for death at all times. So most likely, one of the twins would have died before the other, and the surviving twin would have taken the throne. If both had remained alive, then I think that one of them would have given up his right to his brother.
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But Melkor also was there, and he came to the house of Fëanor, and there he slew Finwë King of the Noldor before his doors, and spilled the first blood in the Blessed Realm; for Finwë alone had not fled from the horror of the Dark.
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