The point is that a lot of CW arguing ignores the dilemma you describe. Slave-holders saw themselves as abiding by the Constitution (and the Declaration), and Abolitionists, driven by a novel interpretation of Christianity, sought to change the historic status quo.
In other words, Liberals were ignoring the clear meaning of Constitutional requirements and substituting their own personal feelings for the existing law. Same as today.
I just yesterday read a statement made by a man at the South Carolina secession convention and he touched on exactly the point you mentioned. He said the South was holding true to the meaning of the Declaration and the US Constitution, but it's requirements were being routinely violated by the Northern states.