As far as what he believed, it seems King was an evangelical Christian which would at least partly contradict your assertion.
Although we are all frauds and sinners, each one of us.
After a cursory glance at King’s “Letter from a Birmingham Jail,” I can’t find anything which contradicts conservative Christian principles—in fact, he says things that show he agreed with the most important ones.
For example:
“An unjust law is a human law that is not rooted in eternal and natural law.”
“As far as what he believed, it seems King was an evangelical Christian “
You need to do better research. King was a heretic who denied the authority of the Scripture, the bodily resurrection of Christ and IIRC the virgin birth.