There are some differences between the Constitutions and I'm not convinced, partially because the vice president of the Confederacy himself seemed to think that it was "unmistakably" protected in the Constitution:

"I congratulate the country that the strife has been put to rest forever, and that American slavery is to stand before the world as it is, and on its own merits. We have now placed our domestic institution, and secured its rights unmistakably, in the Constitution. We have sought by no euphony to hide its name. We have called our negroes 'slaves', and we have recognized and protected them as persons and our rights to them as property."

Edit: The privileges and immunities clause of the US 14th amendment seems to have a parallel in the Confederate Constitution so it's not entirely clear that the Constitutions are the same here (and it seems like if the Constitution didn't protect slavery it was an oversight or someone just forgot to tell their vice president). Apparently the privileges and immunities clause in the US Constitution was essentially nullified later (the US Supreme Court seems to have just wacky interpretations sometimes) but seems intended to confer rights to people in states. I'm a bit out of my depth in finding primary sources on this though, except for the excerpt from the VP who has a very clear opinion (and I have a tendency to not immediately believe what a VP is saying).

Article IV Section 2(1)

The citizens of each State shall be entitled to all the privileges and immunities of citizens in the several States; and shall have the right of transit and sojourn in any State of this Confederacy, with their slaves and other property; and the right of property in said slaves shall not be thereby impaired.