The situation with the US Constitution as opposed to the British one is somewhat different, given that the Confederate constitution was largely modeled upon the US constitution. Given that there is not even a single equivalent document in the United Kingdom for the US to have used as a model in forming its constitution, the comparison strikes me as somewhat out of place.
As far as the fact that the CSA constitution made much over slavery, I would say that this was probably prompted by the fact that slavery was the most important issue in prompting seccession, which is why their constitution was edited to explicitly and repeatedly confirm the right to own slaves.
Note also some of the guarantees given to those who owned human being effectively weakened the concept of States Rights!!