Problem #1, the Confederate Constitution expressly forbade the central government from tampering with slavery. It's right there in Article I: "no... law denying or impairing the right of property in negro slaves shall be passed."
Problem #2, amending the Constitution would be a real bitch. Much harder than amending the US Constitution.
Article V of the Confederate Constitution lays out the process. First, three states have to call special conventions asking for amendment. Then, the Confederate Congress calls a special super-convention of all the states. Then if that convention approves the amendment, it goes back to the states, and two-thirds of them have to approve. Slow, cumbersome, and not easy at all. (N.B., this was a feature, not a bug.)
Okay, so no abolition at the federal level. What about individual states abolishing slavery?
Problem #3, Article IV: "The citizens of each State... 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." IOW, if kentucky abolishes slavery, a slaveowner from Texas could still move there, buy land, and work it with his slaves. So, some disincentive to local abolitionism.
What about new territories... the Confederacy conquering Cuba, say? Article IV again: "In all such territory the institution of negro slavery, as it now exists in the Confederate States, shall be recognized and protected."
IOW, even if Cuba (or wherever) did not have slavery, it would after the Confederates moved in.
So, putting aside economic and social issues, it would be damn'd hard to abolish slavery in the Confederacy just on legal grounds.
Doug M.