Leaving aside the plausibility of getting socialists to unify anywhere, I think that there are three things you need to do to make it work.
Firstly, give American socialism a boost in general. It's easy to become a purist talking shop when you have no chance of affecting things, but if you actually have a chance of influencing politics, it encourages a more cooperative approach. Finding a way to weaken the Red Scare and keep New York from expelling its socialist councillors would be a good start.
Secondly, and closely related, is boosting the influence of radicals in the labour movement as a whole. This would both boost the profile of socialist politics, and offer a potential socialist party a good support base to operate with. Getting Max S. Hayes elected as head of the AFL would be a good start, but that is much easier said than done.
Thirdly, have the second conference of the CPPA vote in favour of creating an independent pro-labour party that can serve absorb the various socialist and radical liberal parties. They then make their big début in 1924, when the Democrats and Republicans both nominated conservatives, allowing them to break onto the political scene, coming in third but with a respectable showing. Come 1929 and the stock market crash, they're in a position to cement themselves as a feature of American politics, they will most likely still be a bit player for the foreseeable future.
An alternative idea I've been having is to have the Democrats split after WW2, with the left-wing forming the nucleus of a centre-left social democratic party, around which the various socialist parties rally. Basically Henry Wallace remains FDR's VP in 1944, and ascends to the Presidency in 1945. In 1948 he wins the Democratic nomination, but refuses to compromise with the right of the party and campaigns on a hard-left platform with a strong civil rights focus. This greatly alienates the right, and even a few moderates in the party, who end up running their own candidates in the South. The Republicans are able to win the election, which reflects poorly on Wallace. Come 1952 the right-wing of the Democrats gain control of the nomination, and refuse to compromise with the left. As a result the left split off and rallies the various non-Communist left-wing parties to their cause. The Democrats shift to the right, whilst the Republicans are able to play the two off each other and hold the centre-ground, becoming the dominant party.