As with so many things during that period, it was a mess, with people, churches, families, generations split.
I must also say that I can only speak from personal experience and so this should be taken as that.
The religious folk supporting the idea of equal rights for Blacks were mostly Black Christians, White Jews and White members of several of what many here have called the Liberal Churches, Lutherans, Methodists, Presbyterians, Episcopalians and Roman Catholics. I also met a few who were Southern Baptist as well as quite a few Friends and one Amish guy I remember since it was the first time he had ever been away from rural PA and he just couldn't stop talking about "How many people there were."
The people opposing rights for Blacks were almost all, 100% White Christians. Not only did the White Christian Churches (including some from the denominations listed as supporting rights for Blacks) oppose equal rights, They preached against it from the pulpit and they actively tried to avoid the results should Blacks actually succeed in gaining rights like riding on the same bus to school as the white kids or go to the same school or live in the same neighborhood or eat in the same resturant.
The period starting around 1957 saw an amazing growth in private Christian Church Schools. These were the "Avoidance Schools" and nearly every conservative church started one. They popped up like toadstools. The sole purpose of those schools was to make sure little white Johnny or little white Janie didn't have to sit next to some monkey. It was not just a southern phenomena either. The same thing was happening in Chicago and Boston and St. Louis and Newark and Hoboken and Philadelphia and Detroit.
Christians supported equal rights for Blacks. Christians also opposed allowing them to have such rights. And Christians were the more violent opposition, not staying in the courts, but bombing churches and schools, killing marchers, terrifying people in their homes, standing on the side walks and calling for the police to "Loose the dogs" or "Turn on the hoses".
It was some of Christianity's Brightest Hours, and some of the Darkest.
Aslan is not a Tame Lion