The difficulty is maintaining a three-party system. If, say, the Laborists surpass the Republicans as viable presidential candidates, the Republicans will be more politically advantaged by joining one of the other parties, and their party will likely decline into obscurity.
Indeed like with the Liberals in OTL Britain, we could now see two big parties the centre left Labor Party and the conservative Democratic Party with a small radical socially and economically liberal and perhaps largely rural/suburban bohemian Republican Party...
IC: "It's a shame Regan accidently killed his gay lover's dog, he's the best president the US never had".