The electoral college system essentially forces a two-party system on the United States.  (The alternative would be proportional representation, as in most parlimentary systems; proportional representation fosters coalitions and hence multiple parties.)  This means that the prospects would not be good for any breakaway from one of the two major parties.  No third party or breakaway faction of a major party has ever won the presidency since the Civil War, though they have on a few occasions (e.g., 2000) decided an outcome as a byproduct.

So I think there can be only one Republican Party (whatever it might call itself).  It may be rational and well-governed or conflict-ridden and chaotic, as it is now, but it will remain, at least in label and general sales pitch, a single political identity.  Sort of like having to get into bed with a porcupine and a polecat.