Not sure if this would be enough to have a full blown Christian Communism, and it might be a bit ASB, but I have an interesting scenario.
IOTL
Norman Thomas and
Huey Long had a public debate in 1934 over the merits of socialism, with Thomas arguing for it and Long against it. Have Thomas decisively win the debate, which causes Long to embrace socialism as being the right path for America (either due to genuine conviction or because he thinks it will be a vote winner) and split from the Democrats to join the Socialist Party. This gives the Socialists a boost to their credibility, especially in Louisiana, and over time they are able to poach other left-leaning populist Democrats dissatisfied with Roosevelt. The Socialist Party grows in popularity, absorbing other minor left-wing and populist parties like that
Farmer-Labour Party and
Progressive Party. During the run up to the 1936 Presidential Norman Thomas stands as the Socialist Party candidate, with Huey Long as his VP. Together they run on a platform of radical wealth redistribution, protectionist tariffs and nationalisation of key industries at home, and a non-interventionist foreign policy.
Long survives the attempt on his life and is badly wounded, but ultimately makes a miraculous recovery. This serves to boost Long's popularity even more, with many seeing it as a miracle, and numerous conspiracy theories abound accusing the government/Democrats/Wall Street of trying to have him killed.
The 1936 elections are hotly contested, with the Democrats winning but with the Socialist Party coming in Second in the Electoral College having won in Wisconsin, Minnesota and Louisiana, as well as respectable showings across the Midwest, in California and major urban areas. The Republicans take a serious beating, only able to win Maine, Vermont and New Hampshire, whilst losing seats in the House and Senate. Many pundits predict that the Republicans are heading towards extinction, a prediction that would be confirmed after the 1938 Congressional elections, where their continued losses lead to their collapse, with the left-wing of the party defecting to the Socialists, and the right and centre flocking to the Democrats as the only party capable of keeping the Socialists out of office. A small number of party loyalists try to keep the Republican Party going, but they ultimately fall into obscurity outside of local elections in New England.
As time goes by the Socialist Party continues to grow. In local elections they build up a stable of Sewer Socialists and begin to make gains in state governments. In the South the Democrats are split between the New Deal left that are trying to steal the Socialists' thunder and right-wing Dixiecrats who feel that the mainstream Democrats have gone too far. This gives the Socialists a fighting chance to win pluralities and later majorities in the South, especially once the African-American population, historically sympathetic to the Republicans, begin to switch their support to the Socialists.
This growing American Socialist movement draws heavily on Christianity and Christian Socialism as a central ideological tenet, in part because of its heavily populist leaning, but also as a means of distinguishing themselves from Communism. Another key factor in their appeal to Christianity is that Long manages to convince Charles Coughlin to support the Socialists (bear in mind that he only really began praising Hitler and Mussolini after Long died so it’s not entirely outside the realm of possibility), presenting them as an alternative to greed-driven Capitalism and Godless Communism, and going a long way to establishing a Socialist Party support base amongst working-class Catholics, in particular Irish immigrants.
That gets you a popular Christian Socialist movement in America. I'm not sure how to get it fully Christian Communist, but it might be possible for elements in the party to become more radical in the 60s and 70s once the Civil Rights and anti-War movements pick up (the religious nature of the Abolitionists and Christian pacifism would be reoccurring motifs in those struggles, and I can see MLK being a card-carrying Socialist Party member). A much stronger mainstream socialist movement might also be in a better position to resist the Second Red Scare, leading to a more radicalised America overall.