Conservative means to conserve/preserve what was, as a general philosophy.
The South was trying to preserve the their rights by keeping things the way they were.
There was a big election, and the guy they didn't like won. So they decided they didn't like the rules of the game anymore and wanted to CHANGE THE RULES.
Changing the rules is revolutionary, not conservative.
The Confederates were attempted revolutionaries, not conservatives.
Moreover this entire “Conservative/Family values/ Christian/ Patriot” drivel is a red herring. Most terms as understood today are hardly even applicable except as anachronism.
The Confederacy was revolutionary, not conservative.
Slavery is not a family value, it demeans interpersonal relationships to one of ownership, and family bonds are broken to serve economic desires.
The South was not more Christian than the North, nor was their cause one that was inherently more Christian.
And the Confederacy were not patriots, but rebels. Their patriotism was to their home State, not to the United States.